Phased matroids are matroids with extra structure. They model vector arrangements over C in the same way that oriented matroids model vector arrangements over R, and many of the topological questions that arise with oriented matroids are at least as intriguing in the context of phased matroids. Just as the theory of oriented matroids leads to a combinatorial model for the real Grassmannians (the "MacPhersonians"), the theory of phased matroids leads to a combinatorial model for complex Grassmannians. I'll discuss the "phased MacPhersonian" and its relation to questions in topology and combinatorics.