  Over the last decade, observations have shown that the mean mass of
ultra-high-energy cosmic rays (UHECRs) increases progressively toward the
highest energies. However, the precise composition is still unknown, and
several theoretical studies hint at the existence of a subdominant proton
component up to the highest energies. Motivated by the exciting prospect of
performing charged-particle astronomy with ultra-high-energy (UHE) protons we
quantify the level of UHE-proton flux that is compatible with present
multimessenger observations and the associated fluxes of neutral messengers
produced in the interactions of the protons. We study this scenario with
numerical simulations of two independent populations of extragalactic sources
and perform a fit to the combined UHECR energy spectrum and composition
observables, constrained by diffuse gamma-ray and neutrino observations. We
find that up to of order $10\%$ of the cosmic rays at the highest energies can
be UHE protons, although the result depends critically on the selected hadronic
interaction model for the air showers. Depending on the maximum proton energy
($E_\text{max}^\text{p}$) and the redshift evolution of sources, the associated
flux of cosmogenic neutrinos and UHE gamma rays can significantly exceed the
multimessenger signal of the mixed-mass cosmic rays. Moreover, if
$E_\text{max}^\text{p}$ is above the GZK limit, we predict a large flux of UHE
neutrinos above EeV energies that is absent in alternate scenarios for the
origin of UHECRs. We present the implications and opportunities afforded by
these UHE proton, neutrino and photon fluxes for future multimessenger
observations.
