  We consider the mixed topological-holomorphic Chern-Simons theory introduced
by Costello, Yamazaki and Witten on a $\mathbb{Z}_2$ orbifold. We use this to
construct semi-classical solutions of the boundary Yang-Baxter equation in the
elliptic and trigonometric cases. A novel feature of the trigonometric case is
that the $\mathbb{Z}_2$ action lifts to the gauge bundle in a $z$-dependent
way. We construct several examples of $K$-matrices, and check they agree with
cases appearing in the literature.
