This paper details the extension of a metric-based anisotropic mesh adaptation strategy to the boundary element method for problems of 3D acoustic wave propagation. Traditional mesh adaptation strategies for boundary element methods rely on Galerkin discretizations of the boundary integral equations, and the development of appropriate error indicators. They often require the solution of further integral equations. These methods utilise the error indicators to mark elements where the error is above a specified tolerance and then refine these elements. Such an approach cannot lead to anisotropic adaptation regardless of how these elements are refined, since the orientation and shape of current elements cannot be modified. In contrast, the method proposed here is independent of the discretization technique (e.g., collocation, Galerkin). Furthermore, it completely remeshes at each refinement step, altering the shape, size, and orientation of each element according to an optimal metric based on a numerically recovered Hessian of the boundary solution. The resulting adaptation procedure is truly anisotropic and independent of the complexity of the geometry. We show via a variety of numerical examples that it recovers optimal convergence rates for domains with geometric singularities. In particular, a faster convergence rate is recovered for scattering problems with complex geometries.