Phased matroids are combinatorial objects, recently defined by Anderson and Delucchi, that play the same role for complex vector spaces as oriented matroids do for real vector spaces. A phased matroid is a matroid with additional structure that generalizes orientation.
According to Mnëv's Universality Theorem, for those phased matroids which are complexified oriented matroids, the realization space can be arbitrarily complicated. In contrast, for most other phased matroids, the realization space is remarkably simple. I will focus on the rank-3 case to demonstrate some properties of, and proofs about, phased matroids.