A head-mounted device may have left-eye and right-eye optical modules that may have a display that creates an image and a corresponding lens that provides the image to an eye box for viewing by a user. The optical modules each include a lens barrel to which the display and lens of that optical module are mounted and a head-mounted optical module gaze tracking system. The gaze tracking system may have light-emitting devices such as light-emitting diodes that emit light at a given near infrared wavelength and may operate through the lens and a removable supplemental lens. The supplemental lens may be a prescription lens and may have an antireflection portion that reduces the reflectance of the lens across visible and near infrared wavelengths. The antireflection portion may be a thin-film interference coating or a moth-eye portion. In this way, the gaze tracking system may operate through the removable supplemental lens.