TITLE: Can $\omega_1$ be supercompact?
QUESTION [9 upvotes]: Is "ZF + $\omega_1$ is supercompact" consistent relative to "ZFC + there is a supercompact cardinal"?
In particular, if $\delta$ is supercompact, does it remain so in $V(\mathbb{R} \cap V[G])$ where $G \subset Col(\omega,<\delta)$ is $V$-generic?  This seems to be the case for measurability but I am having trouble proving it for supercompactness.  It seems likely that someone else has tried this, so I though I'd ask here.
The appropriate definition of supercompactness in ZF is the one in terms of normal fine measures, where normality is defined using diagonal intersections.
I am aware that $\omega_1$ has some amount of supercompactness under AD. I am interested in a more direct proof using forcing, which I hope will give (full) supercompactness.

REPLY [3 votes]: While might not be a full and satisfactory answer, you might be interested in the following paper by Spector:

Spector, M. Iterated extended ultrapowers and supercompactness without choice.,
  Ann. Pure Appl. Logic 54 (1991), no. 2, 179–194.