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