Let $M=G/H$ be a real symmetric space and $𝔤=𝔥+𝔮$ the corresponding decomposition of the Lie algebra. To each open $H$-invariant domain $D_{𝔮}\subseteq i𝔮$ consisting of real ad-diagonalizable elements, we associate a complex manifold $\Xi \left(D_{𝔮}\right)$ which is a curved analog of a tube domain with base $D_{𝔮}$, and we have a natural action of $G$ by holomorphic mappings. We show that $\Xi \left(D_{𝔮}\right)$ is a Stein manifold if and only if $D_{𝔮}$ is convex, that the envelope of holomorphy is schlicht and that $G$-invariant plurisubharmonic functions correspond to convex $H$-invariant...