The discovery of joint sources of high-energy neutrinos and gravitational waves has been a primary target for the LIGO, Virgo, KAGRA, and IceCube observatories. The joint detection of high-energy neutrinos and gravitational waves would provide insight into cosmic processes, from the dynamics of compact object mergers and stellar collapses to the mechanisms driving relativistic outflows. The joint detection of multiple cosmic messengers can also elevate the significance of the common observation even when some or all of the constituent messengers are subthreshold, i.e., not significant enough to declare their detection individually. Using data from the LIGO, Virgo, and IceCube observatories, including subthreshold events, we searched for common sources of gravitational waves and high-energy neutrinos during the third observing run of the Advanced LIGO and Advanced Virgo detectors. Our search did not identify significant joint sources. We derive constraints on the rate densities of joint sources. Our results constrain the isotropic neutrino emission from gravitational-wave sources for very high values of the total energy emitted in neutrinos (>1052–1054 erg).