  The boundary supersymmetric sinh-Gordon model is an integrable quantum field
theory in 1+1 dimensions with bulk N=1 supersymmetry, whose bulk and boundary S
matrices are not diagonal. We present an exact solution of this model. In
particular, we derive an exact inversion identity and the corresponding
thermodynamic Bethe Ansatz equations. We also compute the boundary entropy, and
find a rich pattern of boundary roaming trajectories corresponding to c < 3/2
superconformal models.
