  In the previous paper \cite{Goto_2017}, the notion of an Einstein-Hermitian
metric of a generalized holomorphic vector bundle over a generalized Kahler
manifold of symplectic type was introduced from the moment map framework. In
this paper we establish a Kobayashi-Hitchin correspondence, that is, the
equivalence of the existence of an Einstein-Hermitian metric and
$\psi$-polystability of a generalized holomorphic vector bundle over a compact
generalized Kahler manifold of symplectic type. Poisson modules provide
intriguing generalized holomorphic vector bundles and we obtain $\psi$-stable
Poisson modules over complex surfaces which are not stable in the ordinary
sense.
