Cluster structures on $S{L}_n\mathrm{/}S{O}_n$

We construct cluster structures on the strata ${\mathring{S}}_w$ of a stratification of the symmetric space $S{L}_n\mathrm{/}S{O}_n$ over $\mathbb{C}$. To accomplish this, we study foldings of upper cluster algebras and show that the cluster structures on $S{L}_n\mathrm{/}S{O}_n$ can be obtained from those on $S{L}_n$ via folding. As a corollary, we show that these cluster structures are compatible with the De Concini Poisson structures, and we construct cluster structures on the variety of symmetric matrices ${\text{Sym}}_n$.