We consider a special double scaling limit combining weak coupling and large imaginary twist, for the $\gamma$-twisted N = 4 SYM and establish it also for ABJM theories. The resulting non-gauge chiral 4D and 3D theories of interacting scalars and fermions are integrable in planar limit. In spite of the of the breakdown of conformality by double-trace interactions, most of the correlators for local operators of these theories are conformal, with non-trivial anomalous dimensions, defined by specific Feynman diagrams, which look as regular ``fishnet'' graphs in the bulk, known to be integrable. We discuss the details of this diagrammatics. We construct the doubly scaled asymptotic Bethe ansatz (ABA) equations for multi-magnon states in these theories and show how to use them to compute particular Feynman graphs of $\varphi^4$ theory. These spectral ABA equations fix the diagrams in a given loop order, and the corresponding mizing matrix, up to a few scheme dependent constants, to be fixed from direct computations of the simplest of these graphs. This integrability based method is advocated to be able to compute some high loop order graphs unattainable for other known methods.