Back in the late 90s I invented my own CCG style card game for 40K. There wasn't any Necrons or Tau back then but even a laymen like me could get them all to work perfectly.
I had no over lapping factions. Each faction was on its own.. just like the table top. Then I split the universes other things into a mass of neutral cards linked by type. So.. psyker powers were universal cards, as were thing like orbital bombardment and different maneuvers. The deck building was around mixing tactics NOT factions.
This allows you to have pure.. no lame working together alliance bull between the factions and have in addition to these base units with there activations being in the other cards. 6 factions, to mix with 4 tactic types.