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Abstract The neutrino signal from the next galactic core-collapse supernova will provide an invaluable early warning of the explosion. By combining the burst trigger from several neutrino detectors, the location of the explosion can be triangulated minutes to hours before the optical emission becomes visible, while also reducing the rate of false-positive triggers. To enable multi-messenger follow-up of nearby supernovae, the SuperNova Early Warning System 2.0 (SNEWS 2.0) will produce a combined alert using a global network of neutrino detectors. This paper describes the trigger publishing and alert formation framework of the SNEWS 2.0 network. The framework is built on the HOPSKOTCH publish-subscribe system to easily incorporate new detectors into the network, and it implements a coincidence system to form alerts and estimate a false-positive rate for the combined triggers. The paper outlines the structure of the SNEWS 2.0 software and the initial testing of coincident signals.