This paper deals with the anonymous multi-agent path finding (MAPF) problem for a team of tethered robots. The goal is to find a set of non-crossing paths such that the makespan is minimal. A difficulty comes from the fact that a safety distance must be maintained between two robots when they pass through the same subpath, to avoid collisions and cable entanglements. Hence, robots must be synchronized and waiting times must be added when computing the makespan. We show that bounds can be efficiently computed by solving linear assignment problems. We introduce a variable neighborhood search method to improve upper bounds, and a Constraint Programming model to compute optimal solutions. We experimentally evaluate our approach on three different kinds of instances.