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.
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