  We present a systematic construction of the most general first order
Lagrangian describing an arbitrary number of interacting Maxwell and Proca
fields on Minkowski spacetime. To this aim, we first formalize the notion of a
Proca field, in analogy to the well known Maxwell field. Our definition allows
for a non-linear realization of the Proca mass, in the form of derivative
self-interactions. Consequently, we consider so-called generalized Proca/vector
Galileons. We explicitly demonstrate the ghost-freedom of this complete
Maxwell-Proca theory by obtaining its constraint algebra. We find that, when
multiple Proca fields are present, their interactions must fulfill non-trivial
differential relations in order to ensure the propagation of the correct number
of degrees of freedom. These relations had so far been overlooked, which means
previous multi-Proca proposals generically contain ghosts. This is a companion
paper to arXiv:1905.06968 [hep-th]. It puts on a solid footing the theory there
introduced.
