Just like it says on the tin. How do we prevent Rome from dominating the West as totally as it did from 1054 to the 1500s? Perhaps the Byzantines manage to hold onto Rome for longer, causing a Frankish Empire that doesn't want to accept orders from a Greek puppet to create their own autocephalous church? Maybe if the Arabs conquer Rome, the Pope would be seen as a puppet, much like the Patriarch of Constantinople under Ottoman rule? maybe the Great Schism of the West persists until there are noticeable liturgical differences between nations loyal to Rome and nations loyal to Avignon?
Bonus points if you can get a Patriarch of Canteburry