Antiholomorphic involutions of spherical complex spaces

Let $X$ be a holomorphically separable irreducible reduced complex space, $K$ a connected compact Lie group acting on $X$ by holomorphic transformations, $\theta : K \to K$ a Weyl involution, and $\mu : X \to X$ an antiholomorphic map satisfying $\mu ^2 ={\rm Id}$ and $\mu (kx) = \theta (k)\mu (x)$ for $x\in X, k\in K$. We show that if ${\mathcal O}(X)$ is a multiplicity free $K$-module, then $\mu$ maps every $K$-orbit onto itself. For a spherical affine homogeneous space $X=G/H$ of the reductive group $G=K^{\mathbb{C}}$ we construct an antiholomorphic map $\mu$ with these properties.