Having various wars go even worse for the Ottomans, I could see Christendom taking and holding Constantinople. Then massive arguments erupt between Britain, France, AH and Russia over who gets the Holy City.
Finally, in a diplomatic kludge, a Patriarchal State consisting of Rome, Constantinople, and possible Jerusalem, Antioch and/or Alexandria is formed, under the protection of all the Great Powers, ruled by a council appointed by the respective Patriarchs.
Implausible as all getout, but not QUITE ASB, imo.
With a PoD that late, the only possible plausible rulers of Rome are the Pope and Italy. And no way will Italy get Constantinople. So that leaves the Pope. Of course getting the RC Pope to play nicely with the Orthodox Patriarchs would require a LOT of pressure. That Patriarchal State/Union would probably be unified only on paper....