An idea I've been toying around with recently is that of an ATL analogue to the Investiture Controversy, but early on, in a timeline where Charlemagne's empire stayed united a bit longer (and grew a bit bigger). The increasingly large size of the empire makes any meaningful central control difficult to impossible. Nevertheless, rivalry between the Church and the Emperor does arise over a number of matters. At one point, a particularly devout younger son ascends to the throne unerxpectedly-- after the indended heir dies young in a plague that kills quite a lot of aristocrats. As Emperor, he grants a lot of local fiefs to the Church directly (as Prince-Bishoprics etc.) and allows the Church an inordinate amount of power in decided which aristocrat is entitled to which secular fief. This then causes the exiting tension to boil over, and the next emperor dukes it out with the Pope. The Emperor loses, mainly because the Church has the backing of a large portion of the local aristocrats now. (Not to mention the Church states.)
This then leads to a situation where the Church is dominant in secular politics, and very deliberately keeps all of the (sprawling) Empire decentralised. Makes sense, sinxe centralised secular authority is the only thing that could threaten the primacy of the Pope. To keep potentially troublesome aristocrats out of the way, the Church actively encourages crusades in Eastern Europe, with the undertstanding that the Church will divide the conquered lands into new Imperial fiefs, and select which Crusaders will be granted the titles to those fiefs. (Spoiler: it's the aristocrats who are most loyal to the Church.)
Thus, you'd end up with a Europe that is united under a single Empire, where the secular emperor is a relatively powerless figurehead, where the Church is very powerful, but where temporal authority is intentionally decentralised to a very great degree (and will presumably stay that way for a long time).