  We study a Fefferman-type construction based on the inclusion of Lie groups
${\rm SL}(n+1)$ into ${\rm Spin}(n+1,n+1)$. The construction associates a
split-signature $(n,n)$-conformal spin structure to a projective structure of
dimension $n$. We prove the existence of a canonical pure twistor spinor and a
light-like conformal Killing field on the constructed conformal space. We
obtain a complete characterisation of the constructed conformal spaces in terms
of these solutions to overdetermined equations and an integrability condition
on the Weyl curvature. The Fefferman-type construction presented here can be
understood as an alternative approach to study a conformal version of classical
Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by
the authors. The present work therefore gives a complete exposition of
conformal Patterson-Walker metrics from the viewpoint of parabolic geometry.
