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We introduce some reflection operators on the set of real Borel orbits of the real locus   $X({\mathbb {R}})$   of any spherical complex variety   $X$   defined over   ${\mathbb {R}}$   and homogeneous under a split connected reductive group   $G$   defined also over   ${\mathbb {R}}$  . We thus investigate the existence problem for an action of the Weyl group of   $G$   on the set of real Borel orbits of   $X({\mathbb {R}})$  . In particular, we determine the varieties   $X$   for which the above-mentioned reflection operators define an action of the very little Weyl group of   $X$   on the set of open real Borel orbits of   $X({\mathbb {R}})$  . This enables us to give a parametrization of the   $G({\mathbb {R}})$  -orbits for such   $X({\mathbb {R}})$   in terms of the orbits of this new action.

Real Orbits of Complex Spherical Homogeneous Spaces: the Split Case