The High Altitude Water Cherenkov (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 photo-hadronic or hadronic interactions. The AMON system is running continuously, receiving sub-threshold data (i.e. data that is not suited on its own to do astrophysical searches) from HAWC and IceCube, and combining them in real-time. We present here the analysis algorithm, as well as results from archival data collected between June 2015 and August 2018, with a total live-time of 3.0 years. During this period we found two coincident events that have a false alarm rate (FAR) of $<1$ coincidence per year, consistent with the background expectations. The real-time implementation of the analysis in the AMON system began on November 20th, 2019, and issues alerts to the community through the Gamma-ray Coordinates Network with a FAR threshold of $<4$ coincidences per year.