Many active biological particles, such as swimming microorganisms or motorproteins, do work on their environment by going though a periodic sequence of shapes. Interactions between particles can lead to the phase-synchronization of their duty cycles. We consider collective dynamics in a suspension of such active particles coupled through hydrodynamics. We demonstrate that the emergent non-equilibrium states feature stationary patterned flows and robust unidirectional pumping states under confinement. Moreover the phasesynchronized state of the suspension exhibits spatially robust chimera patterns in which synchronized and phase-isotropic regions coexist within the same system. These findings demonstrate a new route to pattern formation and could guide the design of new active materials. An extension of the same theory for treating ciliated surfaces quantitatively captures the instabilities and flow pumping behaviour of ciliated carpets and metachronal waves.