  We consider scalar local operators of the determinant type in the conformal
``fishnet'' theory that arises as a limit of gamma-deformed $\mathcal{N}=4$
super Yang-Mills theory. We generalise a field-theory approach to expand their
correlation functions to arbitrary order in the small coupling constants and
apply it to the bi-scalar reduction of the model. We explicitly analyse the
two-point functions of determinants, as well as of certain deformations with
the insertion of scalar fields, and describe the Feynman-graph structure of
three- and four-point correlators with single-trace operators. These display
the topology of globe and spiral graphs, which are known to renormalise
single-trace operators, but with ``alternating'' boundary conditions. In the
appendix material we further investigate a four-point function of two
determinants and the shortest bi-local single trace. We resum the diagrams by
the Bethe-Salpeter method and comment on the exchanged OPE states.
