Found it! There's a lookup table that remaps the palette to a darker version of each color.
(Doom called these COLORMAPs)
So when it's trying to draw a dialog over the screen, it looks up each pixel in the COLORMAP instead of just using a 1:1 mapping. Thus, the background darkens, even in a 256-color mode that can't do color blending.
On the Internet, nobody knows you're a dog.