  We study for the first time the effects of strong short-range
electron-electron interactions in generic Rarita--Schwinger--Weyl semimetals
hosting spin-3/2 electrons with linear dispersion at a four-fold band crossing
point. The emergence of this novel quasiparticle, which is absent in
high-energy physics, has recently been confirmed experimentally in the solid
state. We combine symmetry considerations and a perturbative renormalization
group analysis to discern three interacting phases that are prone to emerge in
the strongly correlated regime: The chiral topological semimetal breaks a
$\mathbb{Z}_2$-symmetry and features four Weyl nodes of monopole charge +1
located at vertices of a tetrahedron in momentum space. The s-wave
superconducting state opens a Majorana mass gap for the fermions and is the
leading superconducting instability. The Weyl semimetal phase removes the
fourfold degeneracy and creates two Weyl nodes with either equal or opposite
chirality depending on the anisotropy of the band structure. We find that
symmetry breaking occurs at weaker coupling if the total monopole charge
remains constant across the transition.
