Structured illumination microscopy utilizes illumination of periodic light patterns to allow reconstruction of high spatial frequencies, conventionally doubling the microscope's resolving power. This Letter presents a structured illumination microscopy scheme with the ability to achieve 60 nm resolution by using total internal reflection of a double moiré pattern in high-index materials. We propose a realization that provides dynamic control over relative amplitudes and phases of four coherently interfering beams in gallium phosphide and numerically demonstrate its capability.