[PLEASE SEE EDITS AT BOTTOM OF QUESTION]

Consider the following set-theoretic axiom:

For each set $X$ there exists a set-indexed collection $\{C_i \to X\}_{i\in I_X}$ of surjections such that for every surjection $Z\to X$ there is a map $C_i\to Z$ for some $i$ such that the obvious triangle commutes.

This is known as WISC (Weakly Initial Set of Covers), and can be interpreted as saying Choice fails to hold in at most a 'small' way. It is clearly implied by AC, and I'm willing to bet that it is independent of other usual set-theoretic axioms (ZF, say). WISC is implied by COSHEP (take $I_X$ to be a singleton for all $X$), SVC and AMC.

My questions are these:

- Does anyone know of a weaker choice principle? (Edit: a global choice principle, or at least one for a sizable collection of sets, like all elements $\bigcup_n \mathcal{P}(\mathbb{R}^n)$)

and

- In which popular/common models of set theory does WISC hold? That is, aside from the ones listed at the linked page above (which are particularly category theory-oriented).

(As a bonus question: Come up with a model of ZF that violates WISC or prove we can use forcing to construct one)

Edit: There is also the axiom WISC${}_\kappa$, where we require the set $I_X$ to be bounded by some cardinal $\kappa$ (either less than or at most). This is perhaps more interesting than the unbounded case, especially in topological applications.

Edit2: Benno van den Berg has now shown that Gitik's model of ZF (Israel J. Math 1980) violates WISC. This model, which relies on the consistency of a large cardinal assumption (that is, the existence of an unbounded collection of strongly compact cardinals), has the property that only $\aleph_0$ is a regular cardinal. What Benno showed was that ZF+WISC implies the existence of an unbounded collection of regular cardinals. Now one can clearly ask (thanks to godelian in the comments) whether weaker large cardinal assumptions suffice. One would only need to find a model of ZF in which there is only a bounded collection of regular cardinals. This to me sounds reasonable.