  Let X be a holomorphically separable irreducible reduced complex space, K a
connected compact Lie group acting on X by holomorphic transformations, theta :
K -> K a Weyl involution, and mu : X -> X an antiholomorphic involution map
satisfying mu(kx) = theta(k) mu(x) for x in X and k in K. We show that if the
holomorphic functions on X form 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 (the complexification of K) we construct an antiholomorphic
map mu with these properties.
