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Motivated by Wick-rotations of pseudo-Riemannian manifolds, we study realgeometric invariant theory (GIT) and compatible representations. We extend someof the results from earlier works \cite{W2,W1}, in particular, we give somesufficient as well as necessary conditions for when pseudo-Riemannian manifoldsare Wick-rotatable to other signatures. For arbitrary signatures, we consider aWick-rotatable pseudo-Riemannian manifold with closed $O(p,q)$-orbits, and thusgeneralise the existence condition found in \cite{W1}. Using these existenceconditions we also derive an invariance theorem for Wick-rotations of arbitrarysignatures.

Real GIT with applications to compatible representations and Wick-rotations