Satake diagrams and real structures on spherical varieties

With each antiholomorphic involution $\sigma$ of a connected complex semisimple Lie group $G$ we associate an automorphism $\epsilon_\sigma$ of the Dynkin diagram. The definition of $\epsilon_\sigma$ is given in terms of the Satake diagram of $\sigma$. Let $H \subset G$ be a self-normalizing spherical subgroup. If $\epsilon_\sigma ={\rm id}$ then we prove the uniqueness and existence of a $\sigma$-equivariant real structure on $G/H$ and on the wonderful completion of $G/H$.