  The type III (and the "essential core" type III$_0$) stress-energy tensors in
the Hawking-Ellis (Segre-Plebanski) classification stand out in that there is
to date no known source (either classical or semi-classical) leading to type
III stress-energy. (In contrast the Hawking-Ells types I and II occur
classically, and type IV is known to occur semi-classically). We instead start
by asking the obverse question: What sort of spacetime (assuming the Einstein
equations) needs a type III stress-energy to support it? One key observation is
that type III is incompatible with either planar or spherical symmetry, so one
should be looking at spacetimes of low symmetry (or no symmetry). Finding such
a type III spacetime is a matter of somehow finding an appropriate ansatz for
the metric, calculating the Einstein tensor, and analyzing the pattern of
(Lorentz invariant) eigenvalues and eigenvectors. Herein we report some
(partial) success along these lines - we explicitly exhibit several (somewhat
unnatural) spacetime geometries with a type III Einstein tensor. We then build
an explicit but somewhat odd Lagrangian model leading (in Minkowski space) to
type III stress-energy. While we still have no fully acceptable general
physical model for type III stress-energy, we can at least say something about
what such a stress-energy tensor would entail.
