The GIRM appears to be of two minds about it:

86. While the Priest is receiving the Sacrament, the Communion Chant is begun, its purpose being to express the spiritual union of the communicants by means of the unity of their voices, to show gladness of heart, and to bring out more clearly the “communitarian” character of the procession to receive the Eucharist. The singing is prolonged for as long as the Sacrament is being administered to the faithful.[73] However, if there is to be a hymn after Communion, the Communion Chant should be ended in a timely manner.

Care should be taken that singers, too, can receive Communion with ease.

87. In the Dioceses of the United States of America, there are four options for singing at Communion: (1) the antiphon from the Missal or the antiphon with its Psalm from the Graduale Romanum, as set to music there or in another musical setting; (2) the antiphon with Psalm from the Graduale Simplex of the liturgical time; (3) a chant from another collection of Psalms and antiphons, approved by the Conference of Bishops or the Diocesan Bishop, including Psalms arranged in responsorial or metrical forms; (4) some other suitable liturgical chant (cf. no. 86) approved by the Conference of Bishops or the Diocesan Bishop. This is sung either by the choir alone or by the choir or a cantor with the people.

While the overall purpose of the Communion Chant is to "express the spiritual union of the communicants by the means of the unity of their voices," it simultaneously allows the chant to be sung by the choir alone. My best advice would be to sing a motet based on the text of the communion antiphon assigned for the day (Byrd and Isaac wrote many) with one of the other options to follow if you need more music or to do one of the simplified setting available in the Simple English Propers or other such source with cantor / choir and congregation.

The only thing that is clear about this (and durn'd near ever' other thing 'bout reg-oolations from GIRM) is that the Liturgy Police's jurisdiction extends only as far this forum allows.

Both our Offertory and our Communion tend to go like this:
1) English antiphon
2) Gregorian antiphon (usually only at Communion, but sometimes at the offertory)
3) Motet
4) Congregational hymn