One of Congress' many recent failures is that it has allowed the 2002 Authorization for Use of Military Force (AUMF) to expand to meet the President's every new demand on Iraq. Since the Constitution empowers Congress with declaring war it is perfectly valid to say any use of soldiers in war without one is illegal, yet as a body it doesn't seem to care. This is a tangential issue since we haven't had a formal declaration of war since World War II, but the willingness of Congress to allow war without its explicit approval should at least be mentioned.
The drawback of the half-a-war approach is most obvious now. The AUMF is now being invoked to justify a treaty disguised as a Status of Forces Agreement (SOFA) and it puts Congress in a bind. Of course it is probably as likely that House and Senate leaders are only too happy to let someone else call the shots in Iraq - it looks like it will end up being the biggest foreign policy blunder in our nation's history; no one wants to volunteer to take a piece of it. Embrace it and you have to explain how we continue to turn the mother of all corners and when it will be done. Right now there is not the faintest hint of a policy for Iraq, no vision of what the mission even is at this point. Embracing the war means embracing the man who dictates the plan, and he has none.
Still, if Congress was theoretically interested in getting involved and challenging him on the latest fruits of his seat-of-the-pants strategy there are options. For one it could begin publicly floating drafts of treaties as trial balloons. It would get the word out to Iraq that the body in charge of signing treaties may be willing to collaborate on a document that might lead to a finished product. None of it would be legally binding and the President could disregard it but it might not be easy. We see the same situation in reverse when he presents his "budget" to Congress every year. Someone always says it is dead on arrival and in fact the President has no authority to write the budget or even modify it. His only choices are sign it or veto it.
On the other hand, it still influences the process: It sets parameters for debate by getting major points out in the open. If the President's budget calls for a 25% increase in military spending he probably doesn't expect a 25% increase, but a Congress considering a 5% increase might up it to 7%. It also signals what the President will and won't sign. He could take a hard line and veto anything not entirely to his liking but Americans tend to get cranky when government shuts down on big issues. In other words, Congress could begin to force the President's hand just by getting some ideas out there.
It could also nibble at the SOFA if it wanted. GlobalSecurity.org describes the three kinds as "administrative and technical staff status under the Vienna Convention on Diplomatic Privileges, commonly referred to as A and T status [if they called it technical and administrative they might get more peoples' attention - Dan]; a 'mini' status-of-forces agreement, often used for a short-term presence, such as an exercise; and a full-blown, permanent status-of-forces agreement." Congress could demand that it be classified as one of those three and maybe push for the first (most limited) version. The President would then have to come out in favor of the full-blown version, which would make him look ever more intent on keeping us dug in there. Even with the full-blown version Congress could then say the military had to live up to it and transition to "day-to-day business, such as entry and exit of forces, entry and exit of personal belongings (i.e. automobiles), labor, claims and contractors, and susceptibility to income and sales taxes." In other words, pull back within the country itself. SOFAs are clearly not meant for ongoing combat operations and Congress could get the word out early and often. In doing so they could seize the advantage in the debate.
Once again I realize this is all somewhat academic since there is no evidence Congress would like to take the lead on Iraq. Still, it could begin reclaiming its rightful authority in foreign policy even at this late date. It could begin pulling the President back towards a more traditional executive role and start undoing some of the damage that's been done. There is no reason to expect it will, but the tools are still there, gathering dust.