Filter a view based on an attribute of a user?
Finally found an answer, described in detail here. Confusingly, it starts with creating a view for displaying users. Use a contextual filter to limit it to the currently logged-in user. You can then create a relationship to bring in information about the custom user field, and a second relationship to find nodes that share the value from that field. I tried this and it worked for me!

A more straightforward method is described a few comments earlier, but I haven't tried that one.