  The major simplification in a number of quantum integrable systems is the
existence of special coordinates in which the eigenstates take a factorised
form. Despite many years of studies, the basis realising the separation of
variables (SoV) remains unknown in N=4 SYM and similar models, even though it
is widely believed they are integrable. In this paper we initiate the SoV
approach for observables with nontrivial coupling dependence in a close cousin
of N=4 SYM - the fishnet 4D CFT. We develop the functional SoV formalism in
this theory, which allows us to compute non-perturbatively some nontrivial
observables in a form suitable for numerical evaluation. We present some
applications of these methods. In particular, we discuss the possible SoV
structure of the one-point correlators in presence of a defect, and write down
a SoV-type expression for diagonal OPE coefficients involving an arbitrary
state and the Lagrangian density operator. We believe that many of the findings
of this paper can be applied in the N=4 SYM case, as we speculate in the last
part of the article.
