  In this paper, we consider the inverse scattering problem associated with an
anisotropic medium with a conductive boundary. We will assume that the
corresponding far-field pattern is known/measured and we consider two inverse
problems. First, we show that the far-field data uniquely determines the
boundary coefficient. Next, since it is known that anisotropic coefficients are
not uniquely determined by this data we will develop a qualitative method to
recover the scatterer. To this end, we study the so-called monotonicity method
applied to this inverse shape problem. This method has recently been applied to
some inverse scattering problems but this is the first time it has been applied
to an anisotropic scatterer. This method allows one to recover the scatterer
but considering the eigenvalues of an operator associated with the far--field
operator. We present some simple numerical reconstructions to illustrate our
theory in two dimensions. For our reconstructions, we need to compute the
adjoint of the Herglotz wave function as an operator mapping into $H^1$ of a
small ball.
