  We extend the notion of quasi-transitive orientations of graphs to
2-edge-coloured graphs. By relating quasi-transitive $2$-edge-colourings to an
equivalence relation on the edge set of a graph, we classify those graphs that
admit a quasi-transitive $2$-edge-colouring. As a contrast to
Ghouil\'{a}-Houri's classification of quasi-transitively orientable graphs as
comparability graphs, we find quasi-transitively $2$-edge-colourable graphs do
not admit a forbiddden subgraph characterization. Restricting the problem to
comparability graphs, we show that the family of uniquely quasi-transitively
orientable comparability graphs is exactly the family of comparabilty graphs
that admit no quasi-transitive $2$-edge-colouring.
