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We report multiwavelength observations of the gravitationally lensed blazar QSO B0218+357 in 2016–2020. Optical, X-ray, and GeV flares were detected. The contemporaneous MAGIC observations do not show significant very high energy (VHE; ≳100 GeV) gamma-ray emission. The lack of enhancement in radio emission measured by The Owens Valley Radio Observatory indicates the multizone nature of the emission from this object. We constrain the VHE duty cycle of the source to be <16 2014-like flares per year (95 per cent confidence). For the first time for this source, a broad-band low-state spectral energy distribution is constructed with a deep exposure up to the VHE range. A flux upper limit on the low-state VHE gamma-ray emission of an order of magnitude below that of the 2014 flare is determined. The X-ray data are used to fit the column density of (8.10 ± 0.93stat) × 1021 cm−2 of the dust in the lensing galaxy. VLBI observations show a clear radio core and jet components in both lensed images, yet no significant movement of the components is seen. The radio measurements are used to model the source-lens-observer geometry and determine the magnifications and time delays for both components. The quiescent emission is modelled with the high-energy bump explained as a combination of synchrotron-self-Compton and external Compton emission from a region located outside of the broad-line region. The bulk of the low-energy emission is explained as originating from a tens-of-parsecs scale jet.