  The KM3NeT research infrastructure is under construction in the Mediterranean
Sea. KM3NeT will study atmospheric and astrophysical neutrinos with two
multi-purpose neutrino detectors, ARCA and ORCA, primarily aimed at GeV-PeV
neutrinos. Thanks to the multi-photomultiplier tube design of the digital
optical modules, KM3NeT is capable of detecting the neutrino burst from a
Galactic or near-Galactic core-collapse supernova. This potential is already
exploitable with the first detection units deployed in the sea. This paper
describes the real-time implementation of the supernova neutrino search,
operating on the two KM3NeT detectors since the first months of 2019. A
quasi-online astronomy analysis is introduced to study the time profile of the
detected neutrinos for especially significant events. The mechanism of
generation and distribution of alerts, as well as the integration into the
SNEWS and SNEWS 2.0 global alert systems are described. The approach for the
follow-up of external alerts with a search for a neutrino excess in the
archival data is defined. Finally, an overview of the current detector
capabilities and a report after the first two years of operation are given.
