Motor protein motion on biopolymers can be described by driven diffusive models related to the totally asymmetric exclusion process (TASEP). Inspired by recent experiments on the motion of kinesin-4 motors on antiparallel microtubule overlaps, we develop a model incorporating the TASEP on two antiparallel lanes with binding kinetics and lane switching. We determine the steady-state motor density profiles and phase diagram using phase space flows at the mean-field level. We compared our results to kinetic Monte Carlo simulations. Phases previously identified for the single-filament model occur in the limit of zero switching rate. In addition, lane switching couples the density profiles on the two lanes and can lead to a new phase not possible for a single filament: switching moves motors from the higher density lane to lower density lane, causing local jamming and creating multiple domain walls. We determine the phase diagram of the model for both symmetric and general boundary conditions.