We study the mixed topological/holomorphic Chern-Simons theory of Costello, Witten and Yamazaki on an orbifold (Σ×ℂ)/ℤ2, obtaining a description of lattice integrable systems in the presence of a boundary. By performing an order ℏ calculation we derive a formula for the the asymptotic behaviour of K-matrices associated to rational, quasi-classical R-matrices. The ℤ2-action on Σ × 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.