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Pulsars are remarkable sources, capable of producing EM emission from low-frequency radio waves up to high-energy gamma-rays—and from sites close to the neutron-star surface out to the edges of the magnetosphere. Some pulsars also exhibit several stable "modal" states with fascinating modulation properties and in some cases even different spindown torques. B0943+10 has two such radio modes, one with a carousel of bright accurately drifting subpulses and another with weak chaotic pulses. Correlated mode changes in the radiation at different wavelengths are then key to understanding the physical connections and mechanisms of the emission regions. Through simultaneous observations with the GMRT, LOFAR, and XMM-Newton observatories, we have detected synchronous switching in the radio and X-ray emission properties of B0943+10. When the pulsar is in its radio "bright" mode, the X-rays are unpulsed and have a non-thermal spectrum. Conversely, when the pulsar is in a radio "quiet" mode, the X-ray luminosity more than doubles and a 100%-pulsed thermal component is observed along with the non-thermal radiation. This indicates rapid, global changes in magnetospheric conditions, which challenge all proposed pulsar emission theories.