Unique propulsion systems have evolved in fish that interact with the effects on the surrounding fluid of upstream fish. The downstream fish utilize these complex interactions to swim efficiently. The immersed boundary method is used to explore the phase-mediated locomotion of two self-propelled flexible plates in a tandem arrangement. The interactions caused by the phase difference are elucidated, and the hydrodynamic benefits obtained from the phase-mediated interactions are scrutinized. The variations with the phase difference (Delta phi) and initial gap distance (G(x,0)) in the average cruising speed (UC), the average input power (P), the swimming efficiency (eta), and the equilibrium gap distance (G) are determined. Three flapping modes are identified: a tandem flapping mode, a closely mediated flapping mode, and an interfered flapping mode. The propulsion mechanisms in these modes are analyzed in detail in terms of Delta phi and G(x,0). UC and eta are increased by more than 45% in the closely mediated flapping mode. The vortical structures are visualized to characterize the three flapping modes qualitatively.