  Coupling Dirac structures are Dirac structures defined on the total space of
a fibration, generalizing hamiltonian fibrations from symplectic geometry,
where one replaces the symplectic structure on the fibers by a Poisson
structure.
  We study the associated Poisson gauge theory, in order to describe the
presymplectic groupoid integrating coupling Dirac structures. We find the
obstructions to integrability and we give explicit geometric descriptions of
the integration.
