  We present constraints on the presence of substellar companions to the nearby
($d\sim7.6$ pc), young ($440\pm40$ Myr) K4Ve star TW Piscis Austrini, the wide
($\sim$0.3 pc) companion to the A4V star Fomalhaut. We combined absolute
astrometry from Hipparcos and Gaia with literature radial velocity measurements
and dedicated high-contrast imaging observations obtained with Keck/NIRC2 to
achieve sensitivity to brown dwarf and planetary-mass companions ($\gtrsim2$
M$_{\rm Jup}$) over many decades of orbital period ($\lesssim 10^3$ yr). The
significant astrometric acceleration measured between the Hipparcos and Gaia
catalogues, reported previously in the literature, cannot be explained by the
orbital motion of TW PsA around the barycenter of the Fomalhaut triple system.
Instead, we find that it is consistent with the reflex motion induced by an
orbiting substellar companion. The combination of astrometry, radial
velocities, and a deep $L^{\prime}$ imaging dataset leads to a constraint on
the companion mass of $1.2_{-0.6}^{+0.7}$ M$_{\rm Jup}$. However, the period of
the companion is poorly constrained, with a highly multi-modal period posterior
distribution due to aliasing with the 24.25 year baseline between Hipparcos and
Gaia. If confirmed through continued astrometric or spectroscopic monitoring,
or via direct detection, the companion to TW PsA would represent a choice
target for detailed atmospheric characterization with high-contrast instruments
on the upcoming James Webb Space Telescope and Wide Field Infrared Survey
Telescope.
