It seems like, at this point, string theory is a reasonably well-developed mathematical framework. Tools from string theory are being used to understand things in QCD and condensed matter with no small success.
So what is the obstacle to actually writing down a string theory (using 'string theory' in the most general sense) in $N+1$ dimensions with $N-3$ dimensions compactified in some way? Presumably there is one, but I can't find it discussed in any of the overview papers I've read (and I'm not really in a position to be able to understand anything deeper than an overview paper).
To clarify, I don't mean obstacle in the sense of 'makes predictions we can't yet test', but either mathematical obstacles, or making predictions we already know to be false. I understand that there's a challenge in finding vacuum that has the right value of $\Lambda$, but is this the only thing?
Edit: Just to doubly clarify: it's well known that string theory has a large parameter space. Is finding the correct set of parameters the only thing stopping us from writing down a string theory that describes our universe?