The real Deligne-Mumford moduli space $\bar M_{0,n+1}(\mathbb R)$ of stable genus
zero curves with $n+1$ marked points is a compactification of the space of
configurations of $n$ distinct ordered points on the real line modulo affine
transformations. This space is a smooth projective variety and carries a
natural action of the permutation group $S_n$. We can interpret the
configurations of distinct points on the real line as *diagonal* $n\times n$
matrices with distinct eigenvalues, $S_n$ acting via conjugation with signed
permutation matrices.

**Question**: What is the right compactification of the space of *arbitrary* symmetric $n\times n$ matrices with distinct eigenvalues modulo affine transformations? I am looking for a projective variety carrying a natural $SO(n)$-action, such that if we restrict to diagonal matrices, we recover $\bar M_{0,n+1}(\mathbb R)$ with the above $S_n$-action. Is this variety smooth? Does it have an interpretation in terms of curves with marked points?