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The blazar PKS 0735+178 is possibly associated with multiple neutrino events observed by the IceCube, Baikal, Baksan, and KM3NeT neutrino telescopes while it was flaring in the $γ$-ray, X-ray, ultraviolet and optical bands. We present a detailed study of this peculiar blazar to investigate the temporal and spectral changes in the multi-wavelength emission when the neutrino events were observed. The analysis of Swift-XRT snapshots reveal a flux variability of more than a factor 2 in about $5\times10^3$ seconds during the observation on December 17, 2021. In the $γ$-ray band, the source was in its historical highest flux level at the time of the arrival of the neutrinos. The observational comparison between PKS 0735+178 and other neutrino source candidates, such as TXS 0506+056, PKS 1424+240, and GB6 J1542+6129, shows that all these sources share similar spectral energy distributions, very high radio and $γ$-ray powers, and parsec scale jet properties. Moreover, we present strong supporting evidence for PKS 0735+178 to be, like all the others, a masquerading BL Lac. We perform comprehensive modelling of the multiwavelength emission from PKS 0735+178 within one-zone lepto-hadronic models considering both internal and external photon fields and estimate the expected accompanying neutrino flux. The most optimistic scenario invokes a jet with luminosity close to the Eddington value and the interactions of $\sim$ PeV protons with an external UV photon field. This scenario predicts $\sim 0.067$ muon and antimuon neutrinos over the observed 3-week flare. Our results are consistent with the detection of one very-high-energy neutrino like IceCube-211208A.