An electronic device such as a head-mounted device may have a display that produces a display image. The head-mounted device may have an optical system that merges real-world images from real-world objects with display images. The optical system provides the real-world images and display images to an eye box for viewing by a user. The optical system may use time interleaving techniques and/or polarization effects to merge real-world and display images. Switchable devices such as polarization switches and tunable lenses may be controlled in synchronization with frames of display images. Geometrical phase lenses may be used that exhibit different lens powers to different polarizations of light.