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We present a computational method for reconstructing the refractive index of an unknown complex-shaped two-dimensional medium with embedded metal inclusions from its transverse magnetic electromagnetic scattering properties. We present a novel hybrid surface-volume integral equation that generalises the Lippmann–Schwinger equation to media containing embedded scatterers. Using this hybrid equation, we show that the Frechet derivative of the forward-mapping satisfies a particular inhomogeneous wave scattering problem. We solve the inhomogeneous wave scattering problem using a novel coupled FEM-BEM formulation. Our numerical scheme is based on a thin plate spline ansatz for the refractive index, which can be constructed using a relatively small number of control points, and can be efficiently constructed and evaluated even for the complex-shaped multiply-connected domains of interest. Numerical experiments demonstrate the effectiveness of our method by reconstructing several challenging media. [ABSTRACT FROM AUTHOR]