  We write down an $\mathcal{N}=1$ supersymmetric extension for non-Abelian
gauge theories in (1+3) dimensions with a Lorentz- and CPT-violating term of
the Carroll-Field-Jackiw type. By including effects of the background
(supersymmetric) fermion bilinears that accompany Lorentz-symmetry violation in
a Carroll-Field-Jackiw scenario, we investigate both the gauge boson and
gaugino dispersion relations in order to compute their respective masses in
terms of the background structures. Such results open up a potential path
towards a possible mechanism for gaugino-gauge boson conversion, an analogue of
the Primakoff effect, induced here not by an external magnetic field, but
instead by the (Majorana) fermionic sector of the supersymmetry multiplet in
the backstage of the Lorentz-symmetry violation.
