We use the APOSTLE suite of Local Group (LG) cosmological hydrodynamical simulations to examine the properties of "backsplash" galaxies, i.e, dwarfs outside the virial radius, r 200 , of the Milky Way (MW) or M31 which may have been within their virial boundaries in the past. More than half of all dwarfs at distances between one and two virial radii of each primary are backsplash galaxies. More distant backsplash systems, i.e., those that reach distances well beyond 2 r 200 , are typically systems 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 ∼ 500 kpc of either primary could be a backsplash satellite of either MW or M31. The Tucana dwarf spheroidal (dSph), one of the few known quiescent LG field dwarfs, at d M31 ≈ 1350 kpc and d MW ≈ 880 kpc, is a promising candidate. Tucana's radial velocity suggests it is at rest relative to M31. Further, Tucana is located close to the orbital plane of M33 around M31, and simple orbit integrations indicate that Tucana may have been ejected during an early pericentric passage of M33 about ∼ 11 Gyr ago, a timing which approximately coincides with Tucana's last episode of star formation. We suggest that Tucana may have been a satellite of M31 or M33, providing a compelling explanation for its puzzling lack of gas and ongoing star formation despite its isolated nature.