  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 $\gamma$-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 $\gamma$-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 $\gamma$-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.
