  Various coordinate rings of varieties appearing in the theory of Poisson Lie
groups and Poisson homogeneous spaces belong to the large, axiomatically
defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of
Schubert cells for symmetrizable Kac--Moody groups. We prove that every Poisson
nilpotent algebra satisfying a mild condition on certain scalars is canonically
isomorphic to a cluster algebra which coincides with the corresponding upper
cluster algebra, without additional localizations by frozen variables. The
constructed cluster structure is compatible with the Poisson structure in the
sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are
proved to be equivariant Poisson Unique Factorization Domains. Their seeds are
constructed from sequences of Poisson-prime elements for chains of Poisson
UFDs; mutation matrices are effectively determined from linear systems in terms
of the underlying Poisson structure. Uniqueness, existence, mutation, and other
properties are established for these sequences of Poisson-prime elements.
