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 time-scales. By measuring the cross-correlation function between the IR and X-ray lightcurves, we discovered a significant correlation with an IR 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 $\approx 7.2\,$ 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○.