We investigate Italy's centralized teacher assignment system where teachers can rank "geographical regions", leading to ties in their rank order lists (ROLs). We show that the way ties in teachers' ROLs are resolved in the current assignment mechanism systematically violates teachers' priority rights and results in justified envy. We provide a practical solution to this problem. We introduce a new assignment mechanism called Deferred Acceptance with Hierarchical Choice (DA-HC), which is strategy-proof, eliminates justified envy, and Pareto improves over the benchmark deferred acceptance mechanism with simple tie-breaking (DA-STB). We show that two commonly used efficiency notions from the literature, Pareto efficiency subject to eliminating justified envy and size efficiency subject to eliminating justified envy, do not apply effectively in this context as they are not compatible with strategy-proofness. We introduce a new efficiency concept and show that the DA-HC mechanism is optimal in efficiency terms within the class of strategy-proof mechanisms that eliminate justified envy. Using administrative data, we provide evidence that our proposed mechanism can potentially bring significant welfare improvements over the benchmark DA-STB.