  We report new multi-colour photometry and high-resolution spectroscopic
observations of the long-period variable V501 Aur, previously considered to be
a weak-lined T-Tauri star belonging to the Taurus-Auriga star-forming region.
The spectroscopic observations reveal that V501 Aur is a single-lined
spectroscopic binary system with a 68.8-day orbital period, a slightly
eccentric orbit (e ~ 0.03), and a systemic velocity discrepant from the mean of
Taurus-Auriga. The photometry shows quasi-periodic variations on a different,
~55-day timescale that we attribute to rotational modulation by spots. No
eclipses are seen. The visible object is a rapidly rotating (vsini ~ 25 km/s)
early K star, which along with the rotation period implies it must be large (R
> 26.3 Rsun), as suggested also by spectroscopic estimates indicating a low
surface gravity. The parallax from the Gaia mission and other independent
estimates imply a distance much greater than the Taurus-Auriga region,
consistent with the giant interpretation. Taken together, this evidence
together with a re-evaluation of the LiI~$\lambda$6707 and H$\alpha$ lines
shows that V501 Aur is not a T-Tauri star, but is instead a field binary with a
giant primary far behind the Taurus-Auriga star-forming region. The large mass
function from the spectroscopic orbit and a comparison with stellar evolution
models suggest the secondary may be an early-type main-sequence star.
