  We present a geometric construction and characterization of $2n$-dimensional
split-signature conformal structures endowed with a twistor spinor with
integrable kernel. The construction is regarded as a modification of the
conformal Patterson--Walker metric construction for $n$-dimensional projective
manifolds. The characterization is presented in terms of the twistor spinor and
an integrability condition on the conformal Weyl curvature. We further derive
complete description of Einstein metrics and infinitesimal conformal symmetries
in terms of suitable projective data. Finally, we obtain an explicit
geometrically constructed Fefferman--Graham ambient metric and show vanishing
of $Q$-curvature.
