We show that phase-repulsive coupling eliminates oscillations in a population of synthetic genetic clocks. For this, we propose an experimentally feasible synthetic genetic network that contains phase repulsively coupled repressilators with broken temporal symmetry. As the coupling strength increases, silencing of oscillations is found to occur via the appearance of an inhomogeneous limit cycle, followed by oscillation death. Two types of oscillation death are observed: For lower couplings, the cells cluster in one of two stationary states of protein expression; for larger couplings, all cells end up in a single (stationary) cellular state. Several multistable regimes are observed along this route to oscillation death.