The High Altitude Water Cerenkov (HAWC) and IceCube observatories, through the Astrophysical Multimessenger Observatory Network (AMON) framework, have developed a multimessenger joint search for extragalactic astrophysical sources. This analysis looks for sources that emit both cosmic neutrinos and gamma rays that are produced in photohadronic or hadronic interactions. The AMON system is running continuously, receiving subthreshold data (i.e., data that are not suited on their own to do astrophysical searches) from HAWC and IceCube, and combining them in real time. Here we present the analysis algorithm, as well as results from archival data collected between 2015 June and 2018 August, with a total live time of 3.0 yr. During this period we found two coincident events that have a false-alarm rate (FAR) of <1 coincidence yr(-1), consistent with the background expectations. The real-time implementation of the analysis in the AMON system began on 2019 November 20 and issues alerts to the community through the Gamma-ray Coordinates Network with an FAR threshold of <4 coincidences yr(-1).