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A key goal of the Event Horizon Telescope is to observe the shadow cast by a black hole. Recent simulations have shown that binary black holes, the progenitors of gravitational waves, present shadows with fractal structure. Here we study the binary shadow structure using techniques from nonlinear dynamics, recognizing shadows as exit basins of open Hamiltonian dynamical systems. We apply a recently developed numerical algorithm to demonstrate that parts of the Majumdar-Papapetrou binary black hole shadow exhibit the Wada property: any point of the boundary of one basin is also on the boundary of at least two additional basins. We show that the algorithm successfully distinguishes between the fractal and regular (i.e., nonfractal) parts of the binary shadow. [ABSTRACT FROM AUTHOR]