  We establish the regularity theory for certain critical elliptic systems with
an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary
constraints. As applications, we prove full regularity and smooth estimates at
the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces.
Our methods also lead to the full interior $\epsilon$-regularity and smooth
estimates for weakly Dirac-harmonic maps in all dimensions.
