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.