  We consider three fundamental classes of compact almost homogeneous manifolds
and show that the complements of singular complex orbits in such manifolds are
endowed with plurisubharmonic exhaustions satisfying complex homogeneous
Monge-Amp\`ere equations. This extends to a new family of mixed type examples
various classical results on parabolic spaces and complexifications of
symmetric spaces. Rigidity results on complex spaces modeled on such new
examples are given.
