  We present optical and infrared (IR) light curves of the enshrouded massive
binary NaSt1 (WR 122) with observations from Palomar Gattini-IR (PGIR), the
Zwicky Transient Facility (ZTF), the Katzman Automatic Imaging Telescope
(KAIT), and the All-Sky Automated Survey for Supernovae (ASAS-SN). The optical
and IR light curves span between 2014 July and 2020 Oct., revealing periodic,
sinusoidal variability from NaSt1 with a $P=305.2\pm1.0$ d period. We also
present historical IR light curves taken between 1983 July and 1989 May that
also indicate NaSt1 exhibits long-term IR variability on timescales of
$\sim$decades. Fixed-period sinusoidal fits to the recent optical and IR light
curves show that amplitude of NaSt1's variability is different at different
wavelengths and also reveal significant phase offsets of $\sim18$ d between the
ZTF $r$ and PGIR $J$ light curves.We interpret the $\sim300$ d period of the
observed variability as the orbital period of a binary system in NaSt1.
Assuming a circular orbit and adopting a range of combined stellar mass values
in the range 20-100 M$_\odot$ in NaSt1, we estimate orbital separations of
$\sim2$-4 au. We suggest that the sinusoidal 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 provide an interpretation on the nature of
NaSt1 and speculate that the mass-transfer process may have been triggered by
Roche-lobe overflow (RLOF) during an eruptive phase of a Ofpe/WN9 star. Lastly,
we claim that NaSt1 ceased RLOF mass transfer $\lesssim3400$ yr ago.
