I always thought that Jerusalem should take a page from the Catholics and become a "Holy City" ruled by three appointed heads of the three religions that consider it to be a Holy Place. A sort of triumvirate, independent of any government.
Maybe that could be a solution in Iraq.
Instead of one unity government, a triumvirate of Kurds, Shias, and Sunnis. The good thing about three is that there would be a majority decision, always.
These three rulers could have their own cabinets and consuls, and then the "working bureacracy" could be something like our own career bureaucrats here. They'd be made up of qualified people from all the different ethnic groups.
A Triarchy.