Yes! This is a well-known challenging problem and using a singular value or eigen decomposition to build a sufficiently regularizing prior model is a natural approach up until one has to face those Steifel manifolds.
That said I’m pretty sure the there’s a way to build useful prior models for these problems that doesn’t involve parameterizing rotations (and hence can be written in Stan right now). The bad news is that I’m pretty sure only because a colleague is working on the problem and I’m not sure when any of the results will be sharable!
Apologies for the poor timing.