Well the Bishop of Rome and Patriarch of Constantinople could be forced to relocate so many times by fluctuating, warring empires that they eventually end up in opposite places.
A reversal scenario to that extent has sort of happened in OTL, for Islam. Non-Sunni Islam today is largely practiced in Iran, northern Yemen, central-eastern Turkey, the Caucasus, and Oman. However, in the heyday of the Shi'a-ruled Fatimid Caliphate, they controlled everything west of the Arabian Desert, while the areas east of that (today ~60% non-Sunni) were Sunni-ruled.
Of course it's not a perfect reversal because you have to include all non-Sunnis instead of just Shi'a, and the Fatimids were mainly Shi'a in their ruling class, but it shows that a limited reversal can happen. Then if the empires created solidify maybe it could be completely opposite from OTL.