  We study the mixed topological / holomorphic Chern-Simons theory of Costello,
Witten and Yamazaki on an orbifold $(\Sigma\times{\mathbb C})/{\mathbb Z}_2$,
obtaining a description of lattice integrable systems in the presence of a
boundary. By performing an order $\hbar$ calculation we derive a formula for
the the asymptotic behaviour of $K$-matrices associated to rational,
quasi-classical $R$-matrices. The ${\mathbb Z}_2$-action on $\Sigma\times
{\mathbb C}$ fixes a line $L$, and line operators on $L$ are shown to be
labelled by representations of the twisted Yangian. The OPE of such a line
operator with a Wilson line in the bulk is shown to give the coproduct of the
twisted Yangian. We give the gauge theory realisation of the Sklyanin
determinant and related conditions in the $RTT$ presentation of the boundary
Yang-Baxter equation.
