  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.
