  We present optical and infrared (IR) light curves of NaSt1, also known as
Wolf-Rayet (WR) 122, with observations from Palomar Gattini-IR (PGIR), the
Zwicky Transient Facility (ZTF), the Katzman Automatic Imaging Telescope
(KAIT), the Asteroid Terrestrial-impact Last Alert System (ATLAS), and the
All-Sky Automated Survey for Supernovae (ASAS-SN). We identify a
$P=309.7\pm0.7$ d photometric period from the optical and IR light curves that
reveal periodic, sinusoidal variability between 2014 July and 2021 July. We
also present historical IR light curves taken between 1983 July and 1989 May,
which show variability consistent with the period of the present-day light
curves. In the past, NaSt1 was brighter in the $J$ band with larger variability
amplitudes than the present-day PGIR values, suggesting that NaSt1 exhibits
variability on longer ($\gtrsim$ decade) timescales. Sinusoidal fits to the
recent optical and IR light curves show that the amplitude of NaSt1's
variability differs at various wavelengths and also reveal significant phase
offsets of $17.0\pm2.5$ d between the ZTF $r$ and PGIR $J$ light curves. We
interpret the $310$ d photometric period from NaSt1 as the orbital period of an
enshrouded massive binary. We suggest that the photometric variability of NaSt1
may arise from variations in the line-of-sight optical depth toward
circumstellar optical/IR emitting regions throughout its orbit due to
colliding-wind dust formation. We speculate that past mass transfer in NaSt1
may have been triggered by Roche-lobe overflow (RLOF) during an eruptive phase
of an Ofpe/WN9 star. Lastly, we argue that NaSt1 is no longer undergoing RLOF
mass transfer.
