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The blazar TXS 0506+056 got into the spotlight of the astrophysical communityin September 2017, when a high-energy neutrino detected by IceCube(IceCube-170922A) was associated at the 3 $\sigma$ level to a $\gamma$-rayflare from this source. This multi-messenger photon-neutrino associationremains, as per today, the most significant one ever observed. TXS 0506+056 wasa poorly studied object before the IceCube-170922A event. To bettercharacterize its broad-band emission, we organized a multi-wavelength campaignlasting 16 months (November 2017 to February 2019), covering the radio-band(Mets\"ahovi, OVRO), the optical/UV (ASAS-SN, KVA, REM, Swift/UVOT), the X-rays(Swift/XRT, NuSTAR), the high-energy $\gamma$ rays (Fermi/LAT) and thevery-high-energy (VHE) $\gamma$ rays (MAGIC). In $\gamma$ rays, the behaviourof the source was significantly different from the 2017 one: MAGIC observationsshow the presence of flaring activity during December 2018, while the sourceonly shows an excess at the 4$\sigma$ level during the rest of the campaign (74hours of accumulated exposure); Fermi/LAT observations show several short(days-to-week timescale) flares, different from the long-term brightening of2017. No significant flares are detected at lower energies. The radio lightcurve shows an increasing flux trend, not seen in other wavelengths. We modelthe multi-wavelength spectral energy distributions in a lepto-hadronicscenario, in which the hadronic emission emerges as Bethe-Heitler andpion-decay cascade in the X-rays and VHE $\gamma$ rays. According to the modelpresented here, the December 2018 $\gamma$-ray flare was connected to aneutrino emission that was too brief and not bright enough to be detected bycurrent neutrino instruments.