Over 40 years ago it was suggested that electron loss in the region of the radiation belts that overlaps with the region of high plasma density called the plasmasphere, within four to five Earth radii, arises largely from interaction with an electromagnetic plasma wave called plasmaspheric hiss. This interaction strongly influences the evolution of the radiation belts during a geomagnetic storm, and over the course of many hours to days helps to return the radiation-belt structure to its 'quiet' pre-storm configuration. Observations have shown that the long-term electron-loss rate is consistent with this theory but the temporal and spatial dynamics of the loss process remain to be directly verified. Here we report simultaneous measurements of structured radiation-belt electron losses and the hiss phenomenon that causes the losses. Losses were observed in the form of bremsstrahlung X-rays generated by hiss-scattered electrons colliding with the Earth's atmosphere after removal from the radiation belts. Our results show that changes of up to an order of magnitude in the dynamics of electron loss arising from hiss occur on timescales as short as one to twenty minutes, in association with modulations in plasma density and magnetic field. Furthermore, these loss dynamics are coherent with hiss dynamics on spatial scales comparable to the size of the plasmasphere. This nearly global-scale coherence was not predicted and may affect the short-term evolution of the radiation belts during active times.