  We study gauge transformations of Dirac structures and the relationship
between gauge and Morita equivalences of Poisson manifolds. We describe how the
symplectic structure of a symplectic groupoid is affected by a gauge
transformation of the Poisson structure on its identity section, and prove that
gauge-equivalent integrable Poisson structures are Morita equivalent. As an
example, we study certain generic sets of Poisson structures on Riemann
surfaces: we find complete gauge-equivalence invariants for such structures
which, on the 2-sphere, yield a complete invariant of Morita equivalence.
