  Despite the uncovered association of a high-energy neutrino with the apparent
flaring state of blazar TXS 0506+056 in 2017, the mechanisms leading to
astrophysical particle acceleration and neutrino production are still
uncertain. Recent studies found that when transparent to $\gamma$-rays,
$\gamma$-flaring blazars do not have the opacity for protons to produce
neutrinos. Here we present observational evidence for an alternative
explanation, in which $\gamma$-ray emission is suppressed during efficient
neutrino production. A large proton and target photon density help produce
neutrinos while temporarily suppress the observable $\gamma$-emission due to a
large $\gamma \gamma$ opacity. We show that the Fermi-LAT $\gamma$-flux of
blazar PKS 1502+106 was at a local minimum when IceCube recorded the coincident
high-energy neutrino IC-190730A. Using data from the OVRO 40-meter Telescope,
we find that radio emission from PKS 1502+106 at the time period of the
coincident neutrino IC-190730A was in a high state, in contrast to earlier time
periods when radio and $\gamma$ fluxes are correlated for both low and high
states. This points to an active outflow that is $\gamma$-suppressed at the
time of neutrino production. We find similar local $\gamma$-suppression in
other blazars, including in MAGIC's TeV flux of TXS\,0506+056 and Fermi-LAT's
flux of blazar PKS B1424-418 at the time of coincident IceCube neutrino
detections. Using temporary $\gamma$-suppression, neutrino-blazar coincidence
searches could be substantially more sensitive than previously assumed,
enabling the identification of the origin of IceCube's diffuse neutrino flux
possibly with already existing data.
