Any CSS you put in ~/.config/gtk-4.0/gtk.css will load, you can put an @import statement there to load from another file.

The issue is not the shadows, the popup window itself is shaped. All that black region is supposed to be transparent. To fix with CSS you'd have to get rid of the "arrow" and make the corners square. But that may reduce usability of the popover since it no longer "points" to the parent. Maybe it could be positioned inside the popup somehow.

Thanks, will check the css solution.

You may also want to update your setup or use a compositor like picom if you want any kind of real transparency effects, X11 compositing is about 15 years old now so I think not having it would qualify as "extremely out of date" in software years...

I am not really a fan of transparency and other effects.

The option is there for you if you need it, at some point you may find that apps require it, and trying to hack around it could be impractical.