  We examine multiple ergodic averages of commuting transformations with
polynomial iterates in which the polynomials may be pairwise dependent. In
particular, we show that such averages are controlled by the Gowers-Host-Kra
seminorms whenever the system satisfies some mild ergodicity assumptions.
Combining this result with the general criteria for joint ergodicity
established in our earlier work, we determine a necessary and sufficient
condition under which such averages are jointly ergodic, in the sense that they
converge in the mean to the product of integrals, or weakly jointly ergodic, in
that they converge to the product of conditional expectations. As a corollary,
we deduce a special case of a conjecture by Donoso, Koutsogiannis, and Sun in a
stronger form.
