This is drawn in large part from an exchange of PMs with Snake Featherston.
For some reason, it seems to be part of Hohenstaufen success timelines to also include a successful, revived ERE. This is fine so far as it goes. But it is assumed for some reason that they cooperate and get along and generally like each other.
The problem is that the two states have several reasons to quarrel. In no particular order:
1) Western suspicion of decadent Greeks/Rhomanian suspicion of barbarian Westerners.
2) The Catholic-Orthodox split. This won't be healed over easily. These things just don't heal over easily.
3) There can only be one "One True Roman Empire". If both are claiming that, they will quarrel. Inevitably.
4) The kind of emperors that would forge a HRE as the kind of power seen here are people of great ambition. And correspondingly high views of themselves and their importance to Christendom. See point #3. This is a major problem if not handled properly.
5) No, somehow screwing the Papacy does not address #2. If it was that easy, we'd see a Uniate Church post-Reformation.
6a) Italy might go to the HRE. The ERE has enough other stuff that it can say "Yeah, sure, whatever." to HRE claims to control Italy...for a while. This won't last forever, but maybe there's an agreement on Italy. Or at least as much of agreement as there could be on areas both sides have a reason to want that has nothing to do with legitimate claims and everything to do with rich lands.
6b) But if the HRE is messing with the Levant or Egypt, it is entering into the ERE's idea of what territory is rightfully its. This isn't just an idea of "we want this particular parcel". This is taking territory the ERE has never truly acknowledged losing a claim to. Is it absurd? Not to the ERE, which is what matters.
So my challenge to the various medieval buffs who think two Roman Empires can (and should) coexist. With a POD in the 12th century, create a situation where the ERE and HRE coexist in relative (by the standards of early modern Europe) harmony.