  A signal of high-energy extraterrestrial neutrinos from unknown source(s) was
recently discovered by the IceCube experiment. Neutrinos are always produced
together with gamma-rays, but the gamma-ray flux from extragalactic sources is
suppressed due to attenuation in the intergalactic medium. We report the
discovery of a gamma-ray excess at high Galactic latitudes starting at energies
300 GeV in the data of the Fermi telescope. We show that the multi-TeV
gamma-ray diffuse emission has spectral characteristics at both low and high
Galactic latitudes compatible with those of the IceCube high neutrino signal in
the same sky regions. This suggests that these gamma-rays are the counterpart
of the IceCube neutrino signal, implying that a sizable part of the IceCube
neutrino flux originates from the Milky Way. We argue that the diffuse neutrino
and gamma-ray signal at high Galactic latitudes originates either from
previously unknown nearby cosmic ray "PeVatron" source(s), an extended Galactic
CR halo or from decays of heavy dark matter particles.
