We investigate the differential geometry and topology of four-dimensional

Lorentzian manifolds $(M,g)$ equipped with a real Killing spinor $\varepsilon$,

where $\varepsilon$ is defined as a section of a bundle of irreducible real

Clifford modules satisfying the Killing spinor equation with non-zero real

constant. Such triples $(M,g,\varepsilon)$ are precisely the supersymmetric

configurations of minimal four-dimensional supergravity and necessarily belong

to the class Kundt of space-times, hence we refer to them as supersymmetric

Kundt configurations. We characterize a class of Lorentzian metrics on

$\mathbb{R}^2\times X$, where $X$ is a two-dimensional oriented manifold, to

which every supersymmetric Kundt configuration is locally isometric, proving

that $X$ must be an elementary hyperbolic Riemann surface when equipped with

the natural induced metric. This yields a class of space-times that vastly

generalize the Siklos class of space-times describing gravitational waves in

AdS$_4$. Furthermore, we study the Cauchy problem posed by a real Killing

spinor and we prove that the corresponding evolution problem is equivalent to a

system of differential flow equations, the real Killing spinorial flow

equations, for a family of functions and coframes on any Cauchy hypersurface

$\Sigma\subset M$. Using this formulation, we prove that the evolution flow

defined by a real Killing spinor preserves the Hamiltonian and momentum

constraints of the Einstein equation with negative curvature and is therefore

compatible with the latter. Moreover, we explicitly construct all

left-invariant evolution flows defined by a Killing spinor on a simply

connected three-dimensional Lie group, classifying along the way all solutions

to the corresponding constraint equations, some of which also satisfy the

constraint equations associated to the Einstein condition.