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Clouds and other features in exoplanet and brown dwarf atmospheres cause variations in brightness as they rotate in and out of view. Ground-based instruments reach the high contrasts and small inner working angles needed to monitor these faint companions, but their small fields of view lack simultaneous photometric references to correct for non-astrophysical variations. We present a novel approach for making ground-based light curves of directly imaged companions using high-cadence differential spectrophotometric monitoring, where the simultaneous reference is provided by a double-grating 360○ vector Apodizing Phase Plate (dgvAPP360) coronagraph. The dgvAPP360 enables high-contrast companion detections without blocking the host star, allowing it to be used as a simultaneous reference. To further reduce systematic noise, we emulate exoplanet transmission spectroscopy, where the light is spectrally dispersed and then recombined into white-light flux. We do this by combining the dgvAPP360 with the infrared Arizona Lenslets for Exoplanet Spectroscopy integral field spectrograph on the Large Binocular Telescope Interferometer. To demonstrate, we observed the red companion HD 1160 B (separation ∼780 mas) for one night, and detect $8.8{{\ \rm per\ cent}}$ semi-amplitude sinusoidal variability with an ∼3.24 h period in its detrended white-light curve. We achieve the greatest precision in ground-based high-contrast imaging light curves of sub-arcsecond companions to date, reaching $3.7{{\ \rm per\ cent}}$ precision per 18-min bin. Individual wavelength channels spanning 3.59–3.99 μm further show tentative evidence of increasing variability with wavelength. We find no evidence yet of a systematic noise floor; hence, additional observations can further improve the precision. This is therefore a promising avenue for future work aiming to map storms or find transiting exomoons around giant exoplanets.