Positroids are matroids realizable by real matrices with all nonnegative maximal minors. They partition the ordered matroids into equivalence classes, called positroid envelope classes, by their Grassmann necklaces. We give an explicit graph construction that shows that every positroid envelope class contains a graphic matroid. We prove that a graphic positroid is the unique matroid in its positroid envelope class. Finally, we show that every graphic positroid has an oriented graph representable by a signed incidence matrix with all nonnegative minors.