In this paper, we consider the problem of defining the field strength of abelian potentials in Poisson Electrodynamics. Considering a local symplectic groupoid as a symplectic realization of the Poisson spacetime, it is possible to define several field strengths with different transformation properties. The introduction of gauge invariant momenta allows to find the relations that exist among all of them: in particular, they all vanish simultaneously. A Poisson Chern–Simons model is then proposed and its equations of motion are shortly discussed.