We present the first triangle mesh-based technique for tracking the evolution of general three-dimensional multimaterial interfaces undergoing complex topology changes induced by deformations and collisions. Our core representation is a non-manifold triangle surface mesh with material labels assigned to each half-face to distinguish volumetric regions. We advect the vertices of the mesh in a Lagrangian manner, and employ a complete set of collision-safe mesh improvement and topological operations that track and update material labels. In particular, we develop a unified, collision-safe strategy for handling complex topological operations acting on evolving triple- and higher-valence junctions, and a flexible method to merge colliding multimaterial meshes. We demonstrate our approach with a number of challenging geometric flows, including passive advection, normal flow, and mean curvature flow.