We extend tropicalization and tropical compactification of subvarieties algebraic tori to spherical homogeneous spaces. Given a subvariety, we show that the support colored fan ambient variety agrees with subvariety. The proof is based on our equivariant version flattening by blow-up theorem. provide many examples.

We give a definition of the notion spherical varieties in world complex supervarieties with actions algebraic supergroups. A characterization affine is given which generalizes classical case. also explain some general properties monoid highest weights. Several examples are discussed that interesting their own right and highlight differences case, including regular representation, symmetric supe...