  We use the APOSTLE Local Group (LG) cosmological hydro-simulations to examine
the properties of "backsplash" galaxies, i.e, dwarfs which were within the
virial boundaries of the Milky Way (MW) or M31 in the past, but are today
outside their virial radius ($r_{200}$). More than half of all dwarfs between
$1-2\,r_{200}$ of each primary are backsplash. More distant backsplash systems,
i.e., those reaching distances well beyond $2\,r_{200}$, are typically close to
apocentre of nearly radial orbits, and, therefore, essentially at rest relative
to their primary. We use this result to investigate which LG dwarfs beyond
$\sim500$ kpc of either primary could be a distant backsplash satellite of MW
or M31. Tucana dSph, one of the few known quiescent LG field dwarfs, at $d_{\rm
M31}\approx1350$ kpc and $d_{\rm MW}\approx880$ kpc, is a promising candidate.
Tucana's radial velocity is consistent with being at rest relative to M31.
Further, Tucana is located close to M33's orbital plane around M31, and simple
orbit integrations indicate that Tucana may have been ejected during an early
pericentric passage of M33 $\sim11$ Gyr ago, a timing which approximately
coincides with Tucana's last episode of star formation. We suggest that Tucana
may have been an early-infalling satellite of M31 or M33, providing a
compelling explanation for its puzzling lack of gas and ongoing star formation
despite its isolated nature. In this scenario, M33 should have completed some
orbits around M31, a result that may help to explain the relative dearth of M33
satellite-candidates identified so far.
