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The Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo Interferometer Collaborations have now detected all three classes of compact binary mergers: binary black hole (BBH), binary neutron star (BNS), and neutron star-black hole (NSBH). For coalescences involving neutron stars, the simultaneous observation of gravitational and electromagnetic radiation produced by an event, has broader potential to enhance our understanding of these events, and also to probe the equation of state (EOS) of dense matter. However, electromagnetic follow-up to gravitational wave (GW) events requires rapid real-time detection and classification of GW signals, and conventional detection approaches are computationally prohibitive for the anticipated rate of detection of next-generation GW detectors. In this work, we present the first deep learning based results of classification of GW signals from NSBH mergers in \textit{real} LIGO data. We show for the first time that a deep neural network can successfully distinguish all three classes of compact binary mergers and separate them from detector noise. Specifically, we train a convolutional neural network (CNN) on $\sim 500,000$ data samples of real LIGO noise with injected BBH, BNS, and NSBH GW signals, and we show that our network has high sensitivity and accuracy. Most importantly, we successfully recover the two confirmed NSBH events to-date (GW200105 and GW200115) and the two confirmed BNS mergers to-date (GW170817 and GW190425), together with $\approx 90\%$ of all BBH candidate events from the third Gravitational Wave Transient Catalog, GWTC-3. These results are an important step towards low-latency real-time GW detection, enabling multi-messenger astronomy.