The Orthodox Church considers the celebration of the Divine Liturgy and receipt of the Holy Eucharist the manifestation of the "Body of Christ" among right believers, surrounded by the saints depicted in iconography, through their priest, who are under a right believing, canonical, Orthodox Bishop.
Why the Roman Catholic Church is prepared to accept Eastern Orthodox Christians to its chalice, I do not know, some sort of concession to ecumenical concerns perhaps, but the narrative they typically publish in their Missal does advise Orthodox Christians that they should not approach the Chalice during a Roman Catholic Mass, if so precluded by their bishops.
There is officially a "Dialogue of Love" between the churches, but sharing of the Chalice cannot occur until there is "Unity of the Faith" unified belief, among their respective clergy and faithful. The Orthodox liturgical petitions, "...for the good estate of the Holy Churches of God, and for the unity of them all..." and "Asking for the Unity of the Faith and the Communion of the Holy Spirit..." refer to Orthodox Christians and to their churches, the "Holy Orthodox Churches." The liturgical exclamation, prior to the recitation of the Nicene-Constantinopolin Creed, the "Symbol of Faith," "The doors, the doors, in wisdom let us be attentive," is a reminder to close the doors of the church to non-believers, so that the Orthodox Christians of One Faith participate in the Divine Liturgy, the Eucharistic Service of the Church and its highest form of prayer. "In the fear of God, with faith, and love, draw near," the celebrant's invitation to commune, is another reminder of the importance of the "Faith" the communicants share. And the faith, is the faith of the Undivided Church, and its clergy and faithful who accept the dogmas of the 7 Ecumenical Synods (Councils) of the Early Church, without innovations, including later day "dogma" propounded by a single hierarch, when he sits "Ex-Cathedra." Those who do not share the "faith," are not to be admitted.
The "Dialogue of Love," has produced some understanding of dogmatic differences between the church's, but no reconciliation of them. There is no "Unity of Faith," between the Eastern Orthodox Christian Church and the Roman Catholic Church.
(None of my comments above are intended to be argumentative, though they can be so understood, but to respond to the inquiry of the poster, without necessarily detailing all the differences between the two faiths.)