Characterising tractable fragments of the constraint satisfaction problem (CSP) is an important challenge in theoretical computer science and artificial intelligence. Forbidding patterns (generic subinstances) provides a means of defining CSP fragments which are neither exclusively language-based nor exclusively structure-based. It is known that the class of binary CSP instances in which the broken-triangle pattern (BTP) does not occur, a class which includes all tree-structured instances, are decided by arc consistency (AC), a ubiquitous reduction operation in constraint solvers. We provide a characterisation of irreducible partially-ordered forbidden patterns which have this AC-solvability property. It turns out that BTP is just one of five such AC-solvable patterns. The four other patterns allow us to exhibit new tractable classes.