Sovereignty
If they ask us to leave, we must and we will. If they choose an Islamic Republic over democracy then that will be their choice.
I suspect neither of those things will happen though. Iraqis are enjoying the political freedoms and free press and I doubt they will want to give that up by imposing an Islamic dictatorship. And until they have a standing army and security from both remnants of the former regime and terrorists from the outside I doubt they will ask us to leave.
The new government will be in charge and we will be their guests as long as they want us. Bremer will go home unless he stays on as ambassador.