  We describe all Poisson brackets compatible with the natural cluster algebra
structure in the open Schubert cell of the Grassmannian $G_k(n)$ and show that
any such bracket endows $G_k(n)$ with a structure of a Poisson homogeneous
space with respect to the natural action of $SL_n$ equipped with an R-matrix
Poisson-Lie structure. The corresponding R-matrices belong to the simplest
class in the Belavin-Drinfeld classification. Moreover, every compatible
Poisson structure can be obtained this way.
