For each wall line on screen that is facing the player, I construct either 2 or 3 triangles for the umbra, depending on whether the projection of the player’s position onto the line falls between the endpoints or not. The points used are the actual endpoints of the line and their projection away from the player a fixed distance guaranteed to get them offscreen.

For the soft edges, I similarly project triangles away from the player and texture them with a penumbra texture (making sure not to generate more than 1 triangle in the same place). Note: I tried to just use vertex colours to achieve this, with the outer edges having an aloha value of zero, but I concluded that it is impossible, due to the linear nature of the rasterisation interpolation.

These shadows are of course wrong, the penumbra should be centred on the endpoints, not sticking out from them, but it does the job nicely and makes the darkness even more enveloping.