Making a cross-category search page

I’m working on a page for a legal foundation. They have a “Cases” category covering posts related to cases they’ve covered. Almost all of them also have other categories, for example “Aboriginal Justice” etc. Unfortunately these are all parent categories and not sub-categories.

I want to make a page that will list the Cases posts, but allow the user to narrow down the results by selecting other categories that these posts also share. So basically a search box with a pop-up that only lists the other categories that Cases posts have.

I have experience with PHP and WP, just no idea where to start on this one. Can anyone point me towards the right functions to use?