Synchronization of large spin Hall nano-oscillator (SHNO) arrays is an appealing approach toward ultrafast non-conventional computing. However, interfacing to the array, tuning its individual oscillators and providing built-in memory units remain substantial challenges. Here, we address these challenges using memristive gating of W/CoFeB/MgO/AlOx-based SHNOs. In its high resistance state, the memristor modulates the perpendicular magnetic anisotropy at the CoFeB/MgO interface by the applied electric field. In its low resistance state the memristor adds or subtracts current to the SHNO drive. Both electric field and current control affect the SHNO auto-oscillation mode and frequency, allowing us to reversibly turn on/off mutual synchronization in chains of four SHNOs. We also demonstrate that two individually controlled memristors can be used to tune a four-SHNO chain into differently synchronized states. Memristor gating is therefore an efficient approach to input, tune and store the state of SHNO arrays for non-conventional computing models.