We introduce a Poisson variety compatible with a cluster algebra structure
and a compatible toric action on this variety. We study Poisson and topological
properties of the union of generic orbits of this toric action. In particular,
we compute the number of connected components of the union of generic toric
orbits for cluster algebras over real numbers. As a corollary we compute the
number of connected components of refined open Bruhat cells in Grassmanians
G(k,n) over real numbers.Comment: minor change