  Vectorially structured light has emerged as an enabling tool in many diverse
applications, from communication to imaging, exploiting quantum-like
correlations courtesy of a non-separable spatially varying polarization
structure. Creating these states at the source remains challenging and is
presently limited to two-dimensional vectorial states by customized lasers.
Here we invoke ray-wave duality in a simple laser cavity to produce
polarization marked multi-path modes that are non-separable in three degrees of
freedom and in eight dimensions. As a topical example, we use our laser to
produce the complete set of Greenberger-Horne-Zeilinger (GHZ) basis states,
mimicking high-dimensional multi-partite entanglement with classical light,
which we confirm by a new projection approach. We offer a complete theoretical
framework for our laser based on SU(2) symmetry groups, revealing a rich
parameter space for further exploitation. Our approach requires only a
conventional laser with no special optical elements, is easily scaleable to
higher dimensions, and offers a simple but elegant solution for at-the-source
creation of classically entangled states of structured light, opening new
applications in simulating and enhancing high-dimensional quantum systems.
