We describe a phase-adjusted realification procedure that embeds any finite set of rays in ${\mathbb{C}}^3$ into ${\mathbb{R}}^6$. By assigning an appropriate phase to each ray before applying the standard coordinatewise map, we can arrange that two rays are orthogonal in ${\mathbb{C}}^3$ if and only if their images are orthogonal in ${\mathbb{R}}^6$, 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 ${\mathbb{C}}^3$ Kochen-Specker configuration obtained from mutually unbiased bases, list these 165 rays explicitly in ${\mathbb{C}}^3$, and give for each of them its image in ${\mathbb{R}}^6$ under the canonical realification map. We also note that, because the original three -element contexts are no longer maximal in ${\mathbb{R}}^6$, the embedded configuration admits two-valued states even though its realization with maximal contexts in ${\mathbb{C}}^3$ is Kochen-Specker uncolorable.

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• Quantum correlations, foundations & formalism