This paper addresses the challenge of mutual interference in phase-modulated continuous wave (PMCW) millimeter-wave (mmWave) automotive radar systems. The increasing demand for advanced driver assistance systems (ADAS) has led to a proliferation of vehicles equipped with mmWave radar systems that operate in the same frequency band, resulting in mutual interference that can degrade radar performance creating safety hazards. We consider scenarios involving two similar PMCW radar systems and propose an effective technique for a cooperative design of transmit waveforms such that the mutual interference between them is minimized. The proposed approach is numerically evaluated via simulations of a mmWave automotive radar system. The results demonstrate that the proposed technique notably reduces mutual interference and enhances radar detection performance while imposing very little computational cost and a negligible impact on existing infrastructure in practical automotive radar systems