  We present some advances in the understanding of type III stress-energy
tensors as per the Hawking-Ellis classification. Type I and type II naturally
appear in classical situations, and can also describe semiclassical effects.
Type IV often shows up in semiclassical gravity. Type III is much more subtle.
We focus our attention on type III$_0$ stress-energy tensors, which capture the
essence ("essential core") of type III. Reflecting on known purely
phenomenological examples, ("gyratons"), we are able to generalize the geometry
generated by those type III$_0$ stress-energy tensors. Moreover, we also
succeed in extending work by Griffiths based on massless Weyl spinors by
finding a fundamental classical bosonic Lagrangian description of these type
III$_0$ stress-energy tensors. To the best of our knowledge this is the first
time in the literature that a consistent classical bosonic Lagrangian
formulation for type III$_0$ stress-energy has been found.
