This is nothing new, but it’s a choice-y way of thinking about it. Which is really what I enjoy doing.

**Definition.** Let $V$ be a model of $\ZFC$, and $\PP\in V$ be a notion of forcing. We say that a cardinal $\kappa$ is “colloopsed” by $\PP$ (to $\mu$) if every $V$-generic filter $G$ adds a bijection from $\mu$ onto $\kappa$, but there is an intermediate $N\subseteq V[G]$ satisfying $\ZF$ in which there is no such bijection, but there is one for each $\lambda\lt\kappa$.

This means that $\kappa$ has been collapsed by accident! *Oops!* Or rather, it collapsed just because the axiom of choice is present. If we take $\PP$ to be the (finite support) product of $\operatorname{Col}(\omega,\omega_n)$, then $\aleph_\omega$ is colloopsed, but not collapsed. Namely, by restricting ourselves to the inner model defined by bounded collapses we can easily show that $\aleph_\omega$ is in fact the new $\aleph_1$. This is the Feferman-Levy model (under the assumption that the ground model satisfied $V=L$ anyway).

So from now on, when you apply a Levy collapse argument to a singular cardinal, you don't collapse it, you colloops it. I wonder if there is a nice characterization of colloopsing forcings. But I don't expect that to happen (a man can dream, though).