Multi-layer optical thin films can reflect light to guide it along a hollow tube that they internally line. However, reflecting broadband white light for a range of angles requires many precise thin optical layers, which can be too expensive for large-scale applications such as architectural illumination. Here, we present an alternative configuration that requires only one readily achievable aspect of precision—having a substantially constant cross section, perpendicular to the planned longitudinal propagation direction. This aspect conserves a light ray’s longitudinal directional component, while the ray’s path in the perpendicular direction can be macroscopically chaotic, much like the diffuse reflection characteristic arising from many non-absorbing pigment particles. This macroscopic characteristic is described here as “longitudinally specular” and “transversely diffuse.” This new design overcomes two problems that are found with current prism light guides (PLGs), which are a class of hollow light guides that also have a constant cross section. The PLG has two problems: (1) it requires structured surfaces having near-perfect linear prisms that are much larger than the wavelength of light, wasting optical material and increasing absorption; (2) it only works well with light that has been partially collimated, which limits étendue and requires expensive, inefficient input optics. In this paper, a new, to the best of our knowledge, approach, labeled a “multi-layer prism light guide,” simultaneously addresses both problems. Although the new structure is more complex than a conventional PLG, it may be simpler to manufacture. One potential application is for guiding sunlight into buildings for illumination purposes, without requiring complex sun-tracking solar collection optics.