One thing I recomended to someone recently is the Conflicting Interests System for any type of political faction play.
For example. You have factions 1, 2, and 3.
You also have interests A, B, C, D, E, F, any of which can succeed or fail.
You give each of these factions interests they want to succeed, and interests they want to fail:
Faction 1 wants A, B, and C to succeed, but they want D and E to fail.
Faction 2 wants A, D, and E to succeed, but they want B and F to fail.
Faction 3 wants B, F, and D to succeed, but they want A and C to fail.
And then you, the person running the game, decide when each of these interests are important. Maybe A has to do with passing a law about the restriction of magic--Faction 3 is heavy on the magic use and the other two Factions band together to overwhelm them, but Faction 2 needs to be careful, because very soon, Interest E (certain gentrification platforms) will come into effect and Faction 1 is very against it, so they might want to curry Faction 3's favor...
... and so on and so on. Might get a little complicated, but only if you spread a little too far. The key is making sure every issue is represented (and reviled) equally, and that every Faction has a stance (even if the stance is 'neutrality') regarding every issue.