Uniqueness questions in real algebraic transformation groups

Let G be a compact Lie group and H a closed subgroup. We show that the homogeneous space G/H has a unique structure as a real algebraic G-variety. For a real algebraic H-variety we show that the balanced product G×HX has the structure of a real algebraic G-variety, and this structure is uniquely determined by the structure on X. On the other hand, suppose that M is a closed smooth G-manifold with positive dimensional orbit space M/G. If M has a G-equivariant real algebraic model, then it has an uncountable family of birationally inequivalent such models.