  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 $\pm$ 0.012 in phase, with high pulsed
fraction of 70$-$80%. During the Q mode PSR B0823+26 was not detected in X-rays
(2 $\sigma$ 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 hot spot ($T$$\sim$3.6$\times10^6$ K, $R$$\sim$17
m), a larger cooler concentric ring ($T$$\sim$1.1$\times10^6$ K, $R$$\sim$280
m) and an antipodal hot spot ($T$$\sim$1.1$\times10^6 $ K, $R$$\sim$100 m), for
the angle between the rotation axis and line of sight direction $\sim66^\circ$.
The latter is in conflict with the radio derived value of $(84\pm0.7)^\circ$.
The average X-ray flux within hours-long B-mode intervals varied by a factor
$\pm$20%, 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.
