If a number field $K$ is a Galois extension of $\mathbb{Q}$, and $G = \operatorname{Gal}(K/\mathbb{Q})$, then the class group of $K$ is a $\mathbb{Z}[G]$-module, and since $N = \sum_{g \in G} g$ acts as the norm on ideals, sending every ideal to a principal ideal, the class group is in fact a $\mathbb{Z}[G]/\langle N \rangle$-module.

Roughly speaking, the Cohen-Lenstra heuristics predict that the class group of a totally real Galois number field $K$ should behave as a ``random" finite $\mathbb{Z}[G]/\langle N \rangle$-module modulo a random cyclic submodule.

My question is this: Can a similar kind of statement be made about totally complex number fields $K$ that are Galois over $\mathbb{Q}$? For example, would it simply be a random $\mathbb{Z}[G]/\langle N \rangle$-module, without quotienting out a random cyclic submodule, as in the case for imaginary quadratic fields? Or is that too naive?

Along the same lines, what can be said about the Cohen-Lenstra heuristics for the relative class group of a Galois extension of totally complex number fields?