PSR J1023+0038 is the first millisecond pulsar discovered to pulsate in the visible band; such a detection took place when the pulsar was surrounded by an accretion disk and also showed X-ray pulsations. We report on the first high time resolution observational campaign of this transitional pulsar in the disk state, using simultaneous observations in the optical (TNG, NOT, TJO), X-ray (XMM-Newton, NuSTAR, NICER), infrared (GTC) and UV (Swift) bands. Optical and X-ray pulsations were detected simultaneously in the X-ray high intensity mode in which the source spends $sim$ 70% of the time, and both disappeared in the low mode, indicating a common underlying physical mechanism. In addition, optical and X-ray pulses were emitted within a few km, had similar pulse shape and distribution of the pulsed flux density compatible with a power-law relation $F_{ u} propto u^{-0.7}$ connecting the optical and the 0.3-45 keV X-ray band. Optical pulses were detected also during flares with a pulsed flux reduced by one third with respect to the high mode; the lack of a simultaneous detection of X-ray pulses is compatible with the lower photon statistics. We show that magnetically channeled accretion of plasma onto the surface of the neutron star cannot account for the optical pulsed luminosity ($sim 10^{31}$ erg/s). On the other hand, magnetospheric rotation-powered pulsar emission would require an extremely efficient conversion of spin-down power into pulsed optical and X-ray emission. We then propose that optical and X-ray pulses are instead produced by synchrotron emission from the intrabinary shock that forms where a striped pulsar wind meets the accretion disk, within a few light cylinder radii away, $sim$ 100 km, from the pulsar.
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