A weird, excessive Byzantiwank, eh?
Let's see.....will it perhaps be something like the Franks converting to Arianism, resulting in the position of Arianism remaining a lot stronger, so that the Pope of Rome never becomes more than an appointee of the Byzantine Emperor, so a successful reconciliation with the Patriarch of Alexandria and his many followers follows, leading to a great Byzantine cultural and (indirectly) political expansion into the Red Sea and Sub-Saharan Africa?
And would we then see a Justinian-analogue-on-steroids come along, who conquers North Africa, Italy, and perhaps even Spain all in one go, and then makes sure that the Latin rite Christians remain divided and that the position of the Pope of Rome remains clearly inferior to that of the Patriarch of Constantinople and, of course, the Emperor?