My department is splitting. Should I be on one side or get a joint appointment?

As in comments, for sure avoid getting yourself in the cross-hairs of political fights, as a tenure-track person.

Next, indeed, having mixed allegiances is not a good thing when you come up for tenure.

Possibly "being interdisciplinary" is a good thing, depending entirely on departmental politics. I've seen it go both ways (in my U.S. R1 univ): evidence of not being a real mathematician [sic], or, alternatively, being one of a new breed of mathematician.