I have the feeling that making civil wars to split civs is not a minor thing programmingwise meaning you probably won't see it at least in Civ3. Also, having _your_ civ split would cause a larger outcry than the introducing of the culture flip did.
However, there's a way to have civil wars in Civ3 that are a) not killer events, b) relatively easy to implement, c) balanced against other gameplay, d) but still a (minor) challenge and perhaps fun too.
The Revolution
When you hit that "revolution" button to change your government type your advisor does not warn you for nothing. When you start a revolution that's exactly what you will get.
If you have troops in your capital they are disbanded. Then a big barbarian uprising, their number depending on the difficulty level, happens just next to your capital. These barbarians represent the troops that are still loyal to the Old You. Remember the guy who was in charge a turn before? That was the Old You.
Then it's the job of the Revolutionary Leader, i.e. the You You
, to hunt down all the "traitors to the revolution" so they can be given a proper trial and shot.
Gamewise all that would happen is a simple disbanding of troops and triggering a barbarian upraising. That's already coded there. Just add the revolution as the trigger event.
Having a real revolution would produce a "feel" to what happens, and it would make revolutions cost something even to the religious civs.
And why disband the troops in the capital? An experienced player would of course move them out before starting the revolution, but the idea is that the capital is the place most likely to remain, if not loyal to then at least in the control of the old regime. Also, having your capital defenseless will make the barbs attack it. That's also a civil war thing: the revolutionaries and the loyalists are fighting in the capital probably burning it to the ground in the process. (Change the pop-up texts to reflect this.)
What would also be nice is that the barbarians should be troops appropriate to the age in question so that it wouldn't just be tanks against warriors. Heck, the _old_ government ought to be the one with the tanks!