Starting from a problem in spectral graph theory, I got dragged into a problem in combinatorial matrix theory about constructing $n\times n$ real orthogonal matrices with a specified pattern of zero/non-zero entries. (I can expand on this if required.)

Anyway, I have a construction which appears to work (which I have tested in Maple for some small values of $n$), but to prove it in full generality would require showing that a certain system of simultaneous real quadratic equations has a real solution.

I know very little about algebraic geometry and, well, nothing at all about real algebraic geometry, but I was wondering if there were any existing results in the literature which may help with this---as it stands I'm not even sure where to start looking!