Hyperbolic varieties are a generalization of real-rooted polynomials for varieties of codimension more than one. One prominent example is the image of linear space under coordinate-wise inversion. I will discuss the combinatorial structure of varieties that are hyperbolic with respect to the nonnegative Grassmannian, which is intimately related with the theory of positroids. These varieties generalize multivariate stable polynomials and their tropicalizations are locally subfans of the type-A hyperplane arrangement, in which the maximal cones satisfy a non-crossing condition. This is based on joint work with Felipe Rincón and Josephine Yu.