  We report on multiwavelength target-of-opportunity observations of the blazar
PKS 0735+178, located 2.2$^\circ$ away from the best-fit position of the
IceCube neutrino event IceCube-211208A detected on December 8, 2021. The source
was in a high-flux state in the optical, ultraviolet, X-ray, and GeV gamma-ray
bands around the time of the neutrino event, exhibiting daily variability in
the soft X-ray flux. The X-ray data from Swift-XRT and NuSTAR characterize the
transition between the low-energy and high-energy components of the broadband
spectral energy distribution (SED), and the gamma-ray data from Fermi -LAT,
VERITAS, and H.E.S.S. require a spectral cut-off near 100 GeV. Both X-ray and
gamma-ray measurements provide strong constraints on the leptonic and hadronic
models. We analytically explore a synchrotron self-Compton model, an external
Compton model, and a lepto-hadronic model. Models that are entirely based on
internal photon fields face serious difficulties in matching the observed SED.
The existence of an external photon field in the source would instead explain
the observed gamma-ray spectral cut-off in both leptonic and lepto-hadronic
models and allow a proton jet power that marginally agrees with the Eddington
limit in the lepto-hadronic model. We show a numerical lepto-hadronic model
with external target photons that reproduces the observed SED and is reasonably
consistent with the neutrino event despite requiring a high jet power.
