We explore the possibility of a common origin of ultrahigh energy cosmic rays (UHECRs) and astrophysical neutrinos and further constrain sources which are consistent with this possibility. We show that the common origin hypothesis can be satisfied for the Sibyll2.3c hadronic interaction model but is difficult to accommodate for EPOS-LHC, showing that multimessenger analyses have the power to also constrain hadronic physics beyond LHC energies. We constrain properties of UHECR source environments (and potentially astrophysical neutrino sources), including their photon temperature, gas density, size, magnetic field strength and coherence length, using UHECR and neutrino spectra and composition. Our analysis represents a new type of information on UHECR sources, independent of the mechanism responsible for the UHECR acceleration.