  We develop a new framework for the study of generalized Killing spinors,
where generalized Killing spinor equations, possibly with constraints, can be
formulated equivalently as systems of partial differential equations for a
polyform satisfying algebraic relations in the K\"ahler-Atiyah bundle
constructed by quantizing the exterior algebra bundle of the underlying
manifold. At the core of this framework lies the characterization, which we
develop in detail, of the image of the spinor squaring map of an irreducible
Clifford module $\Sigma$ of real type as a real algebraic variety in the
K\"ahler-Atiyah algebra, which gives necessary and sufficient conditions for a
polyform to be the square of a real spinor. We apply these results to
Lorentzian four-manifolds, obtaining a new description of a real spinor on such
a manifold through a certain distribution of parabolic 2-planes in its
cotangent bundle. We use this result to give global characterizations of real
Killing spinors on Lorentzian four-manifolds and of four-dimensional
supersymmetric configurations of heterotic supergravity. In particular, we find
new families of Einstein and non-Einstein four-dimensional Lorentzian metrics
admitting real Killing spinors, some of which are deformations of the metric of
AdS$_4$ space-time.
