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We consider the strongly nonlinear elliptic Dirichlet problem in a connected bounded domain, overdetermined with the constant Neumann condition $F(\nabla u)=c$ on the boundary. Here $F$ is convex and positively homogeneous of degree 1, and its polar $F^\ast$ represents the anisotropic norm on $\mathbb{R}^n$. We prove that, if this overdetermined boundary value problem admits a solution in a suitable weak sense, then $\Omega$ must be of Wulff shape.