We construct orientations on moduli spaces of pseudoholomorphic quilts with seam conditions in Lagrangian correspondences equipped with relative spin structures and determine the effect of various gluing operations on the orientations. We also invest
igate the behavior of the orientations under composition of Lagrangian correspondences.
We use quilted Floer theory to construct functor-valued invariants of tangles arising from moduli spaces of flat bundles on punctured surfaces. As an application, we show the non-triviality of certain elements in the symplectic mapping class groups o
f moduli spaces of flat bundles on punctured spheres.
