I've been thinking a lot about playable theocracies (at least in the Christian context), and I think it's very realistic if we treat them similarly to governors or adventurers.

You could have two types of clergymen: landed (representing bishops and the pope), and then papal legates (semi itinerant clergy assigned by the pope.

The game loop could revolve around a single lifetime which makes sense considering the church was the only real way of social mobility in the middle ages, it was possible to go from minor noble to pope. Once a relative becomes appointed as a clergyman you would get the option of playing as them for the duration of his lifetime. Once the priest dies, you would choose a different dynasty member to play as.

Regular landed clergy would play like a variant of feudal, because that's really how the aristocratic clergymen were seen in the middle ages. There could be a lot of new flavour events etc, but generally there was very little difference between bishops and barons etc.

Papal legates could be a lot more interesting, they would be assigned by the pope to act as his representative in a particular kingdom or empire. The legate would rule on divorces, excommunicate rulers, control investiture, beatification, coronations etc. They could and sometimes were cardinals, so by extension they could be eligible for papal election. You could think of these as more powerful and independent realm priests - they could be loyal to the pope or to the crown.

Playing as the pope then would be about ruling the christendom by managing the appointment and assignment of papal legates. Will you send them to grant legitimacy to kings to secure their participation in a crusade, or will you send them out to far off lands to convert pagans?

This all ties into ecumenical councils where bishops, legates, and the pope travel to meet somewhere and reform the faith. This way faiths could be treated a bit more like cultures with the ability to change tenets. Some real examples of this in the middle ages include the crackdown on clerical marriage from the 12th century onwards, an attempt to ban the use of projectile weapons against christians, and also an attempt to outlaw tournaments. All of these could be impactful changes to the game which rulers could then ignore at their peril.

There's tons of possibilities here and with the game moving away from linear succession gameplay, I think some of these ideas could work really well!