Well, if your girlfriend says that she believes what the Church teaches after she learns it, then she needs to learn her faith. We cannot receive communion in a Protestant parish, and Protestants cannot receive in a Catholic parish because we are NOT in unity. Communion implies a communion of faith. We do not that communion and thus it is a lie to say that we do. To receive inter-communion, is therefore, a lie.
By the way, the Assemblies of God DOES NOT BELIEVE that their communion is the Real Presence of Christ. They do not have the Real Presence either by faith or by apostolic succession of holy orders.
I'll post the entire section of Canon Law pertaining to the Eucharist since you may not have access to it. It is long, but your girlfriend needs this information. I have highlighted in Green or Red some pertinent passages for your particular situation:
TITLE III: THE BLESSED EUCHARIST
Can. 897 The most venerable sacrament is the blessed Eucharist, in which Christ the Lord himself is contained, offered and received, and by which the Church continually lives and grows. The eucharistic Sacrifice, the memorial of the death and resurrection of the Lord, in which the Sacrifice of the cross is forever perpetuated, is the summit and the source of all worship and Christian life. By means of it the unity of God's people is signified and brought about, and the building up of the body of Christ is perfected. The other sacraments and all the apostolic works of Christ are bound up with, and directed to, the blessed Eucharist.
Can. 898 Christ's faithful are to hold the blessed Eucharist in the highest honor. They should take an active part in the celebration of the most august Sacrifice of the Mass; they should receive the sacrament with great devotion and frequently, and should reverence it with the greatest adoration. In explaining the doctrine of this sacrament, pastors of souls are assiduously to instruct the faithful about their obligation in this regard.
CHAPTER I : THE CELEBRATION OF THE EUCHARIST
Can. 899 ß1 The celebration of the Eucharist is an action of Christ himself and of the Church. In it Christ the Lord, through the ministry of the priest, offers himself, substantially present under the appearances of bread and wine, to God the Father, and gives himself as spiritual nourishment to the faithful who are associated with him in his offering.
ß2 In the eucharistic assembly the people of God are called together under the presidency of the Bishop or of a priest authorized by him, who acts in the person of Christ. All the faithful present, whether clerics or lay people, unite to participate in their own way, according to their various orders and liturgical roles.
ß3 The eucharistic celebration is to be so ordered that all the participants derive from it the many fruits for which Christ the Lord instituted the eucharistic Sacrifice.
ARTICLE 1: THE MINISTER OF THE BLESSED EUCHARIST
Can. 900 ß1 The only minister who, in the person of Christ, can bring into being the sacrament of the Eucharist, is a validly ordained priest.
ß2 Any priest who is not debarred by canon law may lawfully celebrate the Eucharist, provided the provisions of the following canons are observed.
Can. 901 A priest is entitled to offer Mass for anyone, living or dead.
Can. 902 Unless the benefit of Christ's faithful requires or suggests otherwise, priests may concelebrate the Eucharist; they are, however, fully entitled to celebrate the Eucharist individually, but not while a celebration is taking place in the same church or oratory.
Can. 903 A priest is to be permitted to celebrate the Eucharist, even if he is not known to the rector of the church, provided either that he presents commendatory letters, not more than a year old, from his own Ordinary or Superior, or that it can be prudently judged that he is not debarred from celebrating.
Can. 904 Remembering always that in the mystery of the eucharistic Sacrifice the work of redemption is continually being carried out, priests are to celebrate frequently. Indeed, daily celebration is earnestly recommended, because, even if it should not be possible to have the faithful present, it is an action of Christ and of the Church in which priests fulfill their principal role.
Can. 905 ß1 Apart from those cases in which the law allows him to celebrate or concelebrate the Eucharist a number of times on the same day, a priest may not celebrate more than once a day.
ß2 If there is a scarcity of priests, the local Ordinary may for a good reason allow priests to celebrate twice in one day or even, if pastoral need requires it, three times on Sundays or holydays of obligation.
Can. 906 A priest may not celebrate the eucharistic Sacrifice without the participation of at least one of the faithful, unless there is a good and reasonable cause for doing so.
Can. 907 In the celebration of the Eucharist, deacons and lay persons are not permitted to say the prayers, especially the eucharistic prayer, nor to perform the actions which are proper to the celebrating priest.
Can. 908 Catholic priests are forbidden to concelebrate the Eucharist with priests or ministers of Churches or ecclesial communities which are not in full communion with the catholic Church.
Can. 909 A priest is not to omit dutifully to prepare himself by prayer before the celebration of the Eucharist, nor afterwards to omit to make thanksgiving to God.
Can. 910 ß1 The ordinary minister of holy communion is a Bishop, a priest or a deacon.
ß2 The extraordinary minister of holy communion is an acolyte, or another of Christ's faithful deputed in accordance with can. 230 ß3.
Can. 911 ß1 The duty and right to bring the blessed Eucharist to the sick as Viaticum belongs to the parish priest, to assistant priests, to chaplains and, in respect of all who are in the house, to the community Superior in clerical religious institutes or societies of apostolic life.
ß2 In a case of necessity, or with the permission at least presumed of the parish priest, chaplain or Superior, who must subsequently be notified, any priest or other minister of holy communion must do this.
ARTICLE 2: PARTICIPATION IN THE BLESSED EUCHARIST
Can. 912 Any baptized person who is not forbidden by law may and must be admitted to holy communion.
Can. 913 ß1 For holy communion to be administered to children, it is required that they have sufficient knowledge and be accurately prepared, so that according to their capacity they understand what the mystery of Christ means, and are able to receive the Body of the Lord with faith and devotion.
ß2 The blessed Eucharist may, however, be administered to children in danger of death if they can distinguish the Body of Christ from ordinary food and receive communion with reverence.
Can. 914 It is primarily the duty of parents and of those who take their place, as it is the duty of the parish priest, to ensure that children who have reached the use of reason are properly prepared and, having made their sacramental confession, are nourished by this divine food as soon as possible. It is also the duty of the parish priest to see that children who have not reached the use of reason, or whom he has judged to be insufficiently disposed, do not come to holy communion.
Can. 915 Those upon whom the penalty of excommunication or interdict has been imposed or declared, and others who obstinately persist in manifest grave sin, are not to be admitted to holy communion.
Can. 916 Anyone who is conscious of grave sin may not celebrate Mass or receive the Body of the Lord without previously having been to sacramental confession, unless there is a grave reason and there is no opportunity to confess; in this case the person is to remember the obligation to make an act of perfect contrition, which includes the resolve to go to confession as soon as possible.
Can. 917 One who has received the blessed Eucharist may receive it again on the same day only within a eucharistic celebration in which that person participates, without prejudice to the provision of can. 921 ß2.
Can. 918 It is most strongly recommended that the faithful receive holy communion in the course of a eucharistic celebration. If, however, for good reason they ask for it apart from the Mass, it is to be administered to them, observing the liturgical rites.
Can. 919 ß1 Whoever is to receive the blessed Eucharist is to abstain for at least one hour before holy communion from all food and drink, with the sole exception of water and medicine.
ß2 A priest who, on the same day, celebrates the blessed Eucharist twice or three times may consume something before the second or third celebration, even though there is not an hour's interval.
ß3 The elderly and those who are suffering from some illness, as well as those who care for them, may receive the blessed Eucharist even if within the preceding hour they have consumed something.
Can. 920 ß1 Once admitted to the blessed Eucharist, each of the faithful is obliged to receive holy communion at least once a year.
ß2 This precept must be fulfilled during paschal time, unless for a good reason it is fulfilled at another time during the year.
Can. 921 ß1 Christ's faithful who are in danger of death, from whatever cause, are to be strengthened by holy communion as Viaticum.
ß2 Even if they have already received holy communion that same day, it is nevertheless strongly suggested that in danger of death they should communicate again.
ß3 While the danger of death persists, it is recommended that holy communion be administered a number of times, but on separate days.
Can. 922 Holy Viaticum for the sick is not to be unduly delayed. Those who have the care of souls are to take assiduous care that the sick are strengthened by it while they are in full possession of their faculties.
Can. 923 Christ's faithful may participate in the eucharistic Sacrifice and receive holy communion in any catholic rite, without prejudice to the provisions of can. 844.
ARTICLE 3: THE RITES AND CEREMONIES OF THE EUCHARISTIC CELEBRATION
Can. 924 ß1 The most holy Sacrifice of the Eucharist must be celebrated in bread, and in wine to which a small quantity of water is to be added.
ß2 The bread must be wheaten only, and recently made, so that there is no danger of corruption.
ß3 The wine must be natural, made from grapes of the vine, and not corrupt.
Can. 925 Holy communion is to be given under the species of bread alone or, in accordance with the liturgical laws, under both species or, in case of necessity, even under the species of wine alone.
Can. 926 In the eucharistic celebration, in accordance with the ancient tradition of the Latin Church, the priest is to use unleavened bread wherever he celebrates Mass.
Can. 927 It is absolutely wrong, even in urgent and extreme necessity, to consecrate one element without the other, or even to consecrate both outside the eucharistic celebration.
Can. 928 The eucharistic celebration is to be carried out either in the Latin language or in another language, provided the liturgical texts have been lawfully approved.
Can. 929 In celebrating and administering the Eucharist, priests and deacons are to wear the sacred vestments prescribed by the rubrics.
Can. 930 ß1 A priest who is ill or elderly, if he is unable to stand, may celebrate the eucharistic Sacrifice sitting but otherwise observing the liturgical laws; he may not, however, do so in public except by permission of the local Ordinary.
ß2 A priest who is blind or suffering from some other infirmity, may lawfully celebrate the eucharistic Sacrifice by using the text of any approved Mass, with the assistance, if need be, of another priest or deacon or even a properly instructed lay person.
ARTICLE 4: THE TIME AND PLACE OF THE EUCHARISTIC CELEBRATION
Can. 931 The celebration and distribution of the Eucharist may take place on any day and at any hour, except those which are excluded by the liturgical laws.
Can. 932 ß1 The eucharistic celebration is to be carried out in a sacred place, unless in a particular case necessity requires otherwise; in which case the celebration must be in a fitting place.
ß2 The eucharistic Sacrifice must be carried out at an altar that is dedicated or blessed. Outside a sacred place an appropriate table may be used, but always with an altar cloth and a corporal.
Can. 933 For a good reason, with the express permission of the local Ordinary and provided scandal has been eliminated, a priest may celebrate the Eucharist in a place of worship of any Church or ecclesial community which is not in full communion with the catholic Church.
In addition, I would advise reading the following articles from Catholic Answers:
This should get your started in the education of your girlfriend.