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 Astronomy with a Neutrino Telescope and Abyss environmental RESearch (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 KRAg model assuming a 5 PeV per nucleon Galactic cosmic-ray cutoff. No significant excess is found. As a consequence, the limits presented in this Letter start constraining the model parameter space for Galactic cosmic-ray production and transport.