  This paper concerns the reconstruction of a complex-valued anisotropic tensor
$\gamma=\sigma+\i\omega\varepsilon$ from knowledge of several internal magnetic
fields $H$, where $H$ satisfies the anisotropic Maxwell system on a bounded
domain with prescribed boundary conditions. We show that $\gamma$ can be
uniquely reconstructed with a loss of two derivatives from errors in the
acquisition of $H$. A minimum number of 6 such functionals is sufficient to
obtain a local reconstruction of $\gamma$. In the special case where $\gamma$
is close to a scalar tensor, boundary conditions are chosen by means of complex
geometric optics (CGO) solutions. For arbitrary symmetric tensors $\gamma$, a
Runge approximation property is used to obtain partial results. This problem
finds applications in the medical imaging modalities Current Density Imaging
and Magnetic Resonance Electrical Impedance Tomography.
