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 γ-rays, γ-flaring blazars do not have the opacity for protons to produce neutrinos. Here we present observational evidence for an alternative explanation, in which γ-ray emission is suppressed during efficient neutrino production. A large proton and target photon density helps produce neutrinos while temporarily suppressing the observable γ-emission due to a large γγ opacity. We show that the Fermi-LAT γ-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 m 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 γ fluxes are correlated for both low and high states. This points to an active outflow that is γ-suppressed at the time of neutrino production. We find similar local γ-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 γ-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.