The blazar TXS 0506+056 got into the spotlight of the astrophysical community in 2017 September, when a high-energy neutrino detected by IceCube (IceCube-170922A) was associated at the 3σ level with a γ-ray flare from this source. This multi-messenger photon-neutrino association remains, as per today, the most significant association ever observed. TXS 0506+056 was a poorly studied object before the IceCube-170922A event. To better characterize its broadband emission, we organized a multiwavelength campaign lasting 16 months (2017 November to 2019 February), covering the radio band (Metsähovi, OVRO), the optical/UV (ASAS-SN, KVA, REM, Swift/UVOT), the X-rays (Swift/XRT, NuSTAR), the high-energy γ rays (Fermi/LAT), and the very high-energy (VHE) γ rays (MAGIC). In γ rays, the behavior of the source was significantly different from the behavior in 2017: MAGIC observations show the presence of flaring activity during 2018 December, while the source only shows an excess at the 4σ level during the rest of the campaign (74 hr of accumulated exposure); Fermi/LAT observations show several short (on a timescale of days to a week) flares, different from the long-term brightening of 2017. No significant flares are detected at lower energies. The radio light curve shows an increasing flux trend that is not seen in other wavelengths. We model the multiwavelength spectral energy distributions in a lepto-hadronic scenario, in which the hadronic emission emerges as Bethe-Heitler and pion-decay cascade in the X-rays and VHE γ rays. According to the model presented here, the 2018 December γ-ray flare was connected to a neutrino emission that was too brief and not bright enough to be detected by current neutrino instruments.