The existence of diffuse Galactic neutrino production is expected from cosmic ray interactions with Galactic gas and radiation fields. Thus, neutrinos are a unique messenger offering the opportunity to test the products of Galactic cosmic ray interactions up to energies of hundreds of TeV. Here we present a search for this production using ten years of ANTARES track and shower data, as well as seven years of IceCube track data. The data are combined into a joint likelihood test for neutrino emission according to the KRA$_gamma$ model assuming a 5 PeV per nucleon Galactic cosmic ray cutoff. No significant excess is found. As a consequence, the limits presented in this work start constraining the model parameter space for Galactic cosmic ray production and transport.
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