yes, but how are relations with the other factions handled?

It would be a bit silly if the Julii are at war with the Greeks, whilst they are allied to the Brutii for example.
Most logical would be if the Senate determinded 'foreign policy' for all Roman factions, unless one of those factions has 'broken off'.
In gameplay terms however that would mean you couldnt decide for yourself who to go to war with (although convincing the Senate to do what you want etc could be a whole subgame in and of itself...).