in the guide here

The others from here that were identified as wall spiders were brown, and they ate ants. Caterpillar wouldn't be a typical food for them. The question remains, who spun the webbing?

I never knew they ate ants! I've been raising 2 since April, but I never knew they ate ants. I'll have to try giving them some! Mine have been eating tiny Hemiptera and springtails that I get sweeping the grass and clover. Of course, I feed almost all of my spiders the same thing unless they are large, then I try small moths and flies that come to the lights.
This is the kind I'm raising

The one in your thumbnail looks like a pretty mature, perhaps even gravid female...

is a sibling(?) of the 2 I have. There were many living on our hotel balcony and I brought 2 different individuals home from vacation. Maybe it is time to check them for maturity.

Oecobius spins small, flat tent-like webs, if I remember correctly; perhaps the caterpillar got too close? I was wondering, too about the greenish cast of the spider -- until I noticed the color of the caterpillar.

-K

Other Oecobius here make very small webs in architectural corners and wait for ants to enter. I favor crediting the caterpillar with the web. This webbing looks like a greater quantity than Oe. would make at one time.

Disclaimer: I'm not a doctor. More to the point, I'm not an arachnologist.

Possibly, ants attacked the caterpillar as it started to tie itself to the surface. Compare . Maybe the spider was lying in wait for ants.

I don't find any other green Oecobius on BugGuide or Google Images. Would eating caterpillar flesh turn a spider's abdomen green?

Yes, I think you're quite right. As to your question -- only if it was a green caterpillar. :-)

-K

Revisiting this once more, I'm not so sure about the caterpillar spinning the web. I, at least, can't say one way or the other.

though I suppose a spider could come across a caterpillar preparing to make its cocoon.