  We introduce a weak concept of Morita equivalence, in the birational context,
for Poisson modules on complex normal Poisson projective varieties. We show
that Poisson modules, on projective varieties with mild singularities, are
either rationally Morita equivalent to a flat partial holomorphic sheaf, or a
sheaf with a meromorphic flat connection or a co-Higgs sheaf. As an
application, we study the geometry of rank two meromorphic rank two
$\mathfrak{sl}_2$-Poisson modules which can be interpreted as a Poisson
analogous to transversally projective structures for codimension one
holomorphic foliations. Moreover, we describe the geometry of the symplectic
foliation induced by the Poisson connection on the projectivization of the
Poisson module.
