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We revisit the K  -band period–luminosity (P–L) relations of Galactic red supergiants using Gaia Data Release 2 parallaxes and up to 70 yr of photometry from AAVSO and ASAS campaigns. In addition, we examine 206 LMC red supergiants using 50 yr of photometric data from the digitized Harvard Astronomical Plate Collection. We identified periods by computing power spectra and calculated the period–luminosity relations of our samples and compared them with the literature. Newly available data tighten the P–L relations substantially. Identified periods form two groups: one with periods of 300–1000 d, corresponding to pulsations, and another with Long Secondary Periods between 1000 and 8000 d. Among the 48 Galactic objects we find shorter periods in 25 stars and long secondary periods in 23 stars. In the LMC sample we identify 85 and 94 red supergiants with shorter and long secondary periods, respectively. The P–L relation of the Galactic red supergiants is in agreement with the red supergiants in both, the Large Magellanic Cloud and the Andromeda galaxy. We find no clear continuity between the known red giant period–luminosity sequences, and the red supergiant sequences investigated here. [ABSTRACT FROM AUTHOR]