They don't have any sovereignty whatsoever as long as they are a bunch of handpicked tools that doesn't even command all the military forces within their borders. A sovereign country that have forces from an ally within their borders can at least order them to leave if nothing else. They don't need to actually order their allies to leave but they must have the power to do so. The Iraqi "government" does not have this power and thus they are NOT sovereign. It will take years before Iraq regain sovereignty.

What we see in Iraq is just a typical puppet government, consisting of "quislings" or "collaborators" of the type that occupying forces have installed countless times throughout history. The true reason to invade another country are typically to to install such a "government" of handpicked people ready to do your bidding. People that take part in such a "government" are typically despised by their people, hunted by assassins and often rounded up and executed after that true sovereignty has been gained. This government cannot afford to lead Iraq to true democracy since that would expose them to even more dangers that they face today. The survivors will most likely try to flee whats left of Iraq when that happens.

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Whoever fights monsters should see to it that in the process he doesn't become a monster.
Friedrich Nietzsche