Marine ice stream dynamics are sensitive to conditions at the grounding line and basal shear stress; variations in subglacial hydrology have been implicated in ice stream speed-up and shutdown. To investigate the interplay between marine ice stream flow and subglacial hydrology, we couple models of a marine ice stream and a subglacial drainage system. The coupled system evolves dynamically due to a positive feedback between ice flow, heat dissipation at the ice stream bed, and basal lubrication. Our results show that depending on the hydraulic conductivity of the bed, distinct dynamic regimes can be identified. These regimes include steady streaming, hydraulically controlled surges, and thermally controlled oscillations. Periodic fast flow can be initiated by activation waves travelling upstream or downstream or quasi-simultaneously everywhere. Different dynamical regimes are characterised by large differences in grounded ice volume, even under modest changes of grounding line positions. These results imply a strong dependence of marine ice-sheet dynamics on evolving hydrological conditions at the bed and highlight the importance of a better understanding of subglacial hydrology.