Abstract We report on results of a multi-band monitoring campaign from radio to γ rays of the high-redshift flat spectrum radio quasar S5 0836+710 during a high activity period detected by the Large Area Telescope on board the Fermi Gamma-ray Space Telescope. Two major flares were detected, in 2015 August and November. In both episodes, the apparent isotropic γ-ray luminosity exceeds 1050 erg s−1, with a doubling time scale of about 3 hours. The high γ-ray activity may be related to a superluminal knot that emerged from the core in 2015 April at the peak of the radio activity and is moving downstream along the jet. The low variability observed in X-rays may indicate that X-ray emission is produced by the low-energy tail of the same electron population that produces the γ-ray emission. The analysis of full-polarization pc-scale radio observations suggests the presence of a limb-brightened polarization structure at about 1 mas from the core in which a rotation measure gradient with a sign change is observed transverse to the jet direction. These characteristics are consistent with a scenario in which Faraday rotation is produced by a sheath of thermal electrons with a toroidal magnetic field surrounding the emitting jet.