  We report on the first simultaneous high-time resolution X-ray and infrared
(IR) observations of a neutron star low mass X-ray binary in its hard state. We
performed $\approx 2\,$h of simultaneous observations of 4U 1728-34 using
HAWK-I@VLT, XMM-Newton and NuSTAR. The source displayed significant X-ray and
IR variability down to sub-second timescales. By measuring the
cross-correlation function between the infrared and X-ray lightcurves, we
discovered a significant correlation with an infrared lead of $\approx
30-40\,$ms with respect to the X-rays. We analysed the X-ray energy dependence
of the lag, finding a marginal increase towards higher energies. Given the sign
of the lag, we interpret this as possible evidence of Comptonization from
external seed photons. We discuss the origin of the IR seed photons in terms of
cyclo-synchrotron radiation from an extended hot flow. Finally, we also
observed the IR counterpart of a type-I X-ray burst, with a delay of
$\approx7.2\,$s. Although some additional effects may be at play, by assuming
that this lag is due to light travel time between the central object and the
companion star, we find that 4U 1728-34 must have an orbital period longer than
$3\,$h and an inclination higher than 8$^\circ$.
