Most of Earth's glaciers are retreating, but some tidewater glaciers are advancing despite increasing temperatures and contrary to their neighbors. This can be explained by the coupling of ice and sediment dynamics: a shoal forms at the glacier terminus, reducing ice discharge and causing advance towards an unstable configuration followed by abrupt retreat, in a process known as the tidewater glacier cycle. Here we use a numerical model calibrated with observations to show that interactions between ice flow, glacial erosion, and sediment transport drive these cycles, which occur independent of climate variations. Water availability controls cycle period and amplitude, and enhanced melt from future warming could trigger advance even in glaciers that are steady or retreating, complicating interpretations of glacier response to climate change. The resulting shifts in sediment and meltwater delivery from changes in glacier configuration may impact interpretations of marine sediments, fjord geochemistry, and marine ecosystems.