  We introduce bi-fermion fishnet theories, a class of models describing
integrable sectors of four-dimensional gauge theories with non-maximal
supersymmetry. Bi-fermion theories are characterized by a single complex scalar
field and two Weyl fermions interacting only via chiral Yukawa couplings. The
latter generate oriented Feynman diagrams forming hexagonal lattices, whose
fishnet structure signals an underlying integrability that we exploit to
compute anomalous dimensions of BMN-vacuum operators. Furthermore, we
investigate Lunin-Maldacena deformations of $\mathcal{N}=2$ superconformal
field theories with deformation parameter $\gamma$ and prove that bi-fermion
models emerge in the limit of large imaginary $\gamma$ and vanishing 't Hooft
coupling $g$, with $g e^{-i \gamma/2}$ fixed. Finally, we explicitly find
non-trivial conformal fixed points and compute the scaling dimensions of
operators for any $\gamma$ and in presence of double-trace deformations.
