After a lot of digging, I figured out the mistake: I used the “Add new item” option of Visual Studio, but this places the source files at the Intermediate/ProjectFiles, rather than Source/[ModuleName]. This causes all kinds of weird issues.
The fix is to move the files to the right directory.
It would be cool if this behaviour can be fixed, as it’s easy to miss! As a work-around, create files directly in the source directory, then include them using Visual Studio’s “Show All Files” button.