Long secondary periods (LSPs), observed in a third of pulsating red giant stars, are the only unexplained type of large-amplitude stellar variability known at this time. Here we show that this phenomenon is a manifestation of a substellar or stellar companion orbiting the red giant star. Our investigation is based on a sample of about 16,000 well-defined LSP variables detected in the long-term OGLE photometric database of the Milky Way and Magellanic Clouds, combined with the mid-infrared data extracted from the NEOWISE-R archive. From this collection, we selected about 700 objects with stable, large-amplitude, well-sampled infrared light curves and found that about half of them exhibit secondary eclipses, thus presenting an important piece of evidence that the physical mechanism responsible for LSPs is binarity. Namely, the LSP light changes are due to the presence of a dusty cloud orbiting the red giant together with the companion and obscuring the star once per orbit. The secondary eclipses, visible only in the infrared wavelength, occur when the cloud is hidden behind the giant. In this scenario, the low-mass companion is a former planet that has accreted a significant amount of mass from the envelope of its host star and grown into a brown dwarf.
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