Before revision:  The tools we develop will be useful for classifications of other classes of signed graphs, and we mention some follow-up research of this kind; the classifications are surprisingly complex .
After revision:  The tools we develop will be useful for classifications of other classes of signed graphs, and we illustrate this by classifying the complexity of irreflexive signed graphs in which the unicoloured edges form some simple structures, namely paths or cycles. The structure of the signed graphs in the polynomial cases is interesting, suggesting they may constitute a nice class of signed graphs analogous to the so-called bi-arc graphs (which characterized the polynomial cases of list homomorphisms to unsigned graphs) .