We present prompt gamma-ray, early NIR/optical, late optical and X-ray observations of the peculiar GRB 070311 discovered by INTEGRAL, in order to gain clues on the mechanisms responsible for the prompt gamma-ray pulse as well as for the early and late multi-band afterglow of GRB 070311. We fitted with empirical functions the gamma-ray and optical light curves and scaled the result to the late time X-rays. The H-band light curve taken by REM shows two pulses peaking 80 and 140 s after the peak of the gamma-ray burst and possibly accompanied by a faint gamma-ray tail. Remarkably, the late optical and X-ray afterglow underwent a major rebrightening between 3x10^4 and 2x10^5 s after the burst with an X-ray fluence comparable with that of the prompt emission extrapolated in the same band. Notably, the time profile of the late rebrightening can be described as the combination of a time-rescaled version of the prompt gamma-ray pulse and an underlying power law. This result supports a common origin for both prompt and late X-ray/optical afterglow rebrightening of GRB 070311 within the external shock scenario. The main fireball would be responsible for the prompt emission, while a second shell would produce the rebrightening when impacting the leading blastwave in a refreshed shock (abridged).