Compact object mergers which produce both detectable gravitational waves and electromagnetic (EM) emission can provide valuable insights into the neutron star equation of state, the tension in the Hubble constant, and the origin of the r-process elements. However, EM follow-up of gravitational wave sources is complicated by false-positive detections, and the transient nature of the associated EM emission. GWSkyNet-Multi is a machine learning model that attempts facilitate EM follow-up by providing real-time predictions of the source of a gravitational wave detection. The model uses information from Open Public Alerts (OPAs) released by LIGO–Virgo within minutes of a gravitational wave detection. GWSkyNet was introduced in Cabero et al. as a binary classifier and uses the OPA skymaps to classify sources as either astrophysical or as glitches. In this paper, we introduce GWSkyNet-Multi, an extension of GWSkyNet which further distinguishes sources as binary black hole mergers, mergers involving a neutron star, or non-astrophysical glitches. GWSkyNet-Multi is a sequence of three one-versus-all classifiers trained using a class-balanced and physically motivated source mass distribution. Training on this data set, we obtain test set accuracies of 93.7% for binary black hole-versus-all, 94.4% for neutron star-versus-all, and 95.1% for glitch-versus-all. We obtain an overall accuracy of 93.4% using a hierarchical classification scheme. Furthermore, we correctly identify 36 of the 40 gravitational wave detections from the first half of LIGO–Virgo’s third observing run (O3a) and present predictions for O3b sources. As gravitational wave detections increase in number and frequency, GWSkyNet-Multi will be a powerful tool for prioritizing successful EM follow-up.