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 6 . By assigning an appropriate phase to each ray before applying the standard coordinatewise 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 three -element contexts are no longer maximal in R 6 , the embedded configuration admits two-valued states even though its realization with maximal contexts in C 3 is Kochen-Specker uncolorable.