  We find a $(1+1)$-dimensional metric solution for a background hosting
various supersymmetric field theories with a single non-chiral real
supercharge. This supersymmetric background is globally hyperbolic even though
it contains a naked null singularity. In this regard, we show that scalar wave
propagation on the background is well-defined and so the curvature singularity
is a {\it mild} one. Taking inspiration from our previous work, we relate the
field theory on this curved background to some classes of $(1+1)$-dimensional
inhomogeneous field theory in the supersymmetric setup. Utilizing our
supersymmetric background, we elucidate the limitations of canonical
quantization and highlight the conceptual advantages of the algebraic approach
to quantization.
