IIRC, the succession of Patriarchs of Jerusalem continued unbroken after the Muslim conquest and the seat was held by Greeks. After the crusaders took the city, I believe they replaced the Greek patriarch with a Latin one.
Well there you go. You couldn't just give one guy both positions. Form really meant something then. I mean they pretended the Roman Empire was in Germany for hundreds of years after all (I know they claimed Italy too, but you get the point.)
Maybe you could have a situation like this:
The Pope sets it up so that one of his favorites, a nephew (or "nephew" or whatever) becomes Patriarch of Jerusalem and then announces that he recognizes the supremacy of that seat over all other Bishoprics and Patriarchies in Christendom.
The case for Jerusalem is probably a bit stronger amongst the Orthodox than Rome ever was, and they have more influence over the new office, so that schism won't likely last long.
However Western Europe now operates with a much freer hand and starts to alter the church fundamentally region-by-region.
Basically just what Chengar said.
But another possibility is Italian trade is more encouraged in an effort to keep a firm hand on at least the Mediterranean basin.
This may lead to bigger and better galleys requiring more slaves. With the Pope no longer frightened of Norman power (but with Greece now potentially in communion with the Catholics again) maybe the raiding in North Africa picks up and they go full time into the slaving business. Permanent (or longer lasting) Norman inroads into the North African coast?
And what would it mean to the Reconquista? Is it weaker or independent of the church, or does the fight to take the peninsula act as a rallying point for Papal authority in an otherwise distant land?