  We address the question of perturbative consistency in the scalar fishnet
models presented by Caetano, Gurdogan and Kazakov\cite{Gurdogan:2015csr,
Caetano:2016ydc}. We argue that their 3-dimensional $\phi^{6}$ fishnet model
becomes perturbatively stable under renormalization in the large $N$ limit, in
contrast to what happens in their 4-dimensional $\phi^{4}$ fishnet model, in
which double trace terms are known to be generated by the RG flow. We point out
that there is a direct way to modify this second theory that protects it from
such corrections. Additionally, we observe that the 6-dimensional $\phi^{3}$
Lagrangian that spans an hexagonal integrable scalar fishnet is consistent at
the perturbative level as well. The nontriviality and simplicity of this last
model is illustrated by computing the anomalous dimensions of its
$\text{tr}\phi_i \phi_j$ operators to all perturbative orders.
