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Despite the spectacular discovery of an astrophysical neutrino flux by IceCube in 2013, its origin remains a mystery. Whatever its sources, we expect the neutrino flux to be accompanied by a comparable gamma-ray flux. These photons should be degraded in energy by electromagnetic cascades and contribute to the diffuse GeV-TeV flux precisely measured by the Fermi-LAT. Population studies have also permitted to identify the main classes of contributors to this flux, which at the same time have not been associated with major neutrino sources in cross-correlation studies. These considerations allow one to set constraints on the origin and spectrum of the IceCube flux, in particular its low-energy part. We find that, even accounting for known systematic errors, the Fermi-LAT data exclude to at least 95% C.L. any extragalactic transparent source class, irrespective of its redshift evolution, if the neutrino spectrum extends to the TeV scale or below. If the neutrino spectrum has an abrupt cutoff at $\sim10$ TeV, barely compatible with current observations, the tension can be reduced, but this way out requires a significant modification to the current understanding of the origin of the diffuse extragalactic gamma-ray flux at GeV energies. In contrast, these considerations do not apply if a sizable fraction of IceCube data originates within the Galactic halo (a scenario however typically in tension with other constraints) or from a yet unidentified class of "opaque" extragalactic emitters, which do not let the high-energy gamma rays get out.