A configuration of $n$ weighted points on $\mathbb{C}P^1$ is called \emph{stable} if the sum of the weights of equal points is strictly less than half the total weight. The moduli space in this case is defined as the stable configurations modulo Mobius transformations.
I would like to know if there is a nice description of the real locus of this moduli space. I think it should be (diffeomorphic to) stable configurations on $S^1 = \mathbb{R}P^1$ modulo the diagonal action of $PSL(2, \mathbb{R})$. But I am not sure how to prove it.