Simultaneous observations of PSR B0823+26 with ESA’s XMM–Newton, the Giant Metrewave Radio Telescope and international stations of the Low Frequency Array revealed synchronous X-ray/radio switching between a radio-bright (B) mode and a radio-quiet (Q) mode. During the B mode we detected PSR B0823+26 in 0.2 − 2 keV X-rays and discovered pulsed emission with a broad sinusoidal pulse, lagging the radio main pulse by 0.208 ± 0.012 in phase, with high pulsed fraction of 70 − 80 per cent. During the Q mode PSR B0823+26 was not detected in X-rays (2σ upper limit a factor ∼9 below the B-mode flux). The total X-ray spectrum, pulse profile and pulsed fraction can globally be reproduced with a magnetized partially ionized hydrogen atmosphere model with three emission components: a primary small hotspot (T ∼ 3.6 × 10^6 K, R ∼17 m), a larger cooler concentric ring (T ∼ 1.1 × 10^6 K, R ∼280 m) and an antipodal hotspot (T ∼ 1.1 × 10^6 K, R ∼100 m), for the angle between the rotation axis and line of sight direction ∼66°. The latter is in conflict with the radio derived value of (84 ± 0.7)°. The average X-ray flux within hours-long B-mode intervals varied by a factor ±20 per cent, possibly correlated with variations in the frequency and lengths of short radio nulls or short durations of weak emission. The correlated X-ray/radio moding of PSR B0823+26 is compared with the anti-correlated moding of PSR B0943+10, and the lack of X-ray moding of PSR B1822−09. We speculate that the X-ray/radio switches of PSR B0823+26 are due to variations in the rate of accretion of material from the interstellar medium through which it is passing.