Three Party Politics
Suppose the Tea Party representatives respond to that situation by forming their own caucus, as they well may, and functioning as an independent body, a virtual third party. In organizing the House, they will presumably sell their support to the orthodox Republicans in exchange for a positions for some of their members. But in future legislative struggles, matters may not be that simple.
Considered as a three party game, and ignoring divisions with the Democratic party, the logic is simple. Obama and the Democrats cannot pass legislation. Neither can the Republicans. Neither can the (hypothetical) Tea Party caucus. Neither can the Republicans allied with the Tea Party. But an alliance of Obama with either of the other two provides majorities in both houses; that plus the President's signature is, short of a successful filibuster, sufficient. And it is not entirely clear that the Republicans, absent the support of Tea Party senators, could mount a filibuster.
It is tempting, but boring, to assume that since the Tea Party is in some sense on the opposite side from the Democrats, no alliances are possible. Reason to reject that assumption is provided by recent British political history. The Liberal Democrats, by most views, were positioned to the left of a Labor party that had become increasingly centrist in the years since Maggie Thatcher's successful Tory rule. But when an election produced a majority for neither of the major parties, it was the Conservatives, not Labor, that they ended up in coalition with. So far, that coalition seems to working.
I admit, however, that I have had a hard time thinking up plausible issues on which the Tea Party and the Democrats might align against the Republicans. It would be easier if the Tea party were more consistently libertarian—one could imagine, for instance, an alliance to pass a federal medical marijuana law, supposing that it occurs to Obama that scaling back the War on Drugs is at this point a more popular policy than strengthening it. But that one strikes me as implausible and nothing else at the moment occurs to me.