I think it all depends on what you mean by "running" Iraq. If al Qaeda took over the reins of government in Baghdad, you'd have an Afghan-like situation. They couldn't "run" Basra and they couldn't "run" the Kurdish north. Depending on how they accommodate to the tribes of the Sunni Triangle, they'd have more or less influence there. You'd have a crazy-quilt situation of local warlords, private armies etc. all on some kind of spectrum of al Qaeda alignment, ranging from open armed opposition to actual al Qaeda fiefdoms. Pretty much the way Israel wants it.
The likelier scenario is that one faction will emerge victorious over the others in a military struggle and within that faction, a small group, probably a triad, of officers will emerge as the leaders, and from that triad, a few years further down the road, one guy will come out as top dog, much as Saddam did years ago.