We study the class of reach-avoid dynamic games in which multiple agents interact noncooperatively, and each wishes to satisfy a distinct target criterion while avoiding a failure criterion. Reach-avoid games are commonly used to express safety-critical optimal control problems found in mobile robot motion planning. Here, we focus on finding time-consistent solutions, in which future motion plans remain optimal even when a robot diverges from the plan early on due to, e.g., intrinsic dynamic uncertainty or extrinsic environment disturbances. Our main contribution is a computationally-efficient algorithm for multi-agent reach-avoid games which renders time-consistent solutions for all players. We demonstrate our approach in two- and three-player simulated driving scenarios, in which our method provides safe control strategies for all agents.