Structured illumination in single-molecule localization microscopy provides new information on the position of molecules and thus improves the localization precision compared to standard localization methods. Here, we used a time-shifted sinusoidal excitation pattern to modulate the fluorescence signal of the molecules whose position information is carried by the phase and recovered by synchronous demodulation. We designed two flexible fast demodulation systems located upstream of the camera, allowing us to overcome the limiting camera acquisition frequency and thus to maximize the collection of photons in the demodulation process. The temporally modulated fluorescence signal was then sampled synchronously on the same image, repeatedly during acquisition. This microscopy, called ModLoc, allows us to experimentally improve the localization precision by a factor of 2.4 in one direction, compared to classical Gaussian fitting methods. A temporal study and an experimental demonstration both show that the short lifetimes of the molecules in blinking regimes impose a modulation frequency in the kilohertz range, which is beyond the reach of current cameras. A demodulation system operating at these frequencies would thus be necessary to take full advantage of this new localization approach. This article is part of the Theo Murphy meeting issue 'Super-resolution structured illumination microscopy (part 2)'.