Faithful real embedding of a three-dimensional complex Kochen-Specker configuration

We describe a phase-adjusted realification procedure that embeds any finite set of rays in C^3 into R^3. By assigning an appropriate phase to each ray before applying the standard coordinate-wise map, we can arrange that two rays are orthogonal in C^3 if and only if their images are orthogonal in R^6, so the construction yields a faithful orthogonal representation of the original complex configuration. As a concrete example, we consider the 165 projectively distinct rays used in a C^3 Kochen-Specker configuration obtained from mutually unbiased bases, list these 165 rays explicitly in C^3, and give for each of them its image in R^6 under the canonical realification map. We also note that, because the original 3-element contexts are no longer maximal in R^6, the embedded configuration admits two-valued states even though its realisation with maximal contexts in C^3 is Kochen-Specker uncolourable.