A max-plus system with partial synchronization is a persistent discrete event system divided into a main subsystem and a secondary subsystem where some events in the secondary subsystem can only occur when (not after) associated events in the main subsystem occur. A formal model based on max-plus recursive equations is presented for max-plus systems with partial synchronization. Furthermore, the control problem of output tracking is addressed by assigning a higher priority to the main subsystem: the performance of the main subsystem is never degraded to improve the performance of the secondary subsystem. Two different control approaches are investigated: optimal feedforward control and model predictive control. In both cases, residuation theory is applied to efficiently solve the optimization problem.