  The galaxy SDSS1335+0728, which had exhibited no prior optical variations
during the preceding two decades, began showing significant nuclear variability
in the Zwicky Transient Facility (ZTF) alert stream from December 2019 (as
ZTF19acnskyy). Its behaviour suggests that SDSS1335+0728 hosts a $\sim 10^6
M_{\odot}$ black hole (BH) that is currently in the process of `turning on'. We
present a multi-wavelength photometric analysis and spectroscopic follow-up
performed with the aim of better understanding the origin of the nuclear
variations detected in SDSS1335+0728. We used archival photometry and
spectroscopic data to study the state of SDSS1335+0728 prior to December 2019,
and new observations from Swift, SOAR/Goodman, VLT/X-shooter, and Keck/LRIS
taken after its turn-on to characterise its current state. We find that: (a)
since 2021, the UV flux is four times brighter than the flux reported by GALEX
in 2004; (b) since June 2022, the mid-infrared flux has risen more than two
times, and the W1-W2 WISE colour has become redder; (c) since February 2024,
the source has begun showing X-ray emission; (d) the narrow emission line
ratios are now consistent with a more energetic ionising continuum; (e) broad
emission lines are not detected; and (f) the [OIII] line increased its flux
$\sim 3.6$ years after the first ZTF alert, which implies a relatively compact
narrow-line-emitting region. We conclude that the variations observed in
SDSS1335+0728 could be either explained by an AGN that is just turning on or by
an exotic tidal disruption event (TDE). If the former is true, SDSS1335+0728 is
one of the strongest cases of an AGN observed in the process of activating. If
the latter, it would correspond to the longest and faintest TDE ever observed
(or another class of still unknown nuclear transient). Future observations of
SDSS1335+0728 are crucial to further understand its behaviour.
