  After proving the impossibility of consistent non-minimal coupling of a real
Rarita-Schwinger gauge field to electromagnetism, we re-derive the necessity of
introducing the graviton in order to couple a complex Rarita-Schwinger gauge
field to electromagnetism, with or without a cosmological term, thereby
obtaining ${\cal N}=2$ pure supergravity as the only possibility. These results
are obtained with the BRST-BV deformation method around the flat and (A)dS
backgrounds in 4 dimensions. The same method applied to $n_{v}$ vectors, ${\cal
N}$ real spin-3/2 gauge fields and at most one real spinor field also requires
gravity and yields ${\cal N}=3$ pure supergravity as well as ${\cal N}=1$ pure
supergravity coupled to a vector supermultiplet, with or without cosmological
terms. Independently from the matter content, we finally derive strong
necessary quadratic constraints on the possible gaugings for an arbitrary
number of spin-1 and spin-3/2 gauge fields, that are relevant for larger
supergravities.
