In OTL, the Muslims quickly came to conquer regions that were home to the major patriarchal sees. Jerusalem, Antioch, Alexandria, and, eventually Constantinople. The Patriarchs continued in their duties as the leaders of their respective Churches.
Your challenge is for a similar situation to occur: A Christian Empire comes to conquer a region where a Muslim Caliph (however real in their authority they may be) is seated. The Caliph should maintain some semblance of religious authority under the rule of this empire.
POD: Any time after, say, AD 700.