Abstract Schlenker (2013) gives a number of puzzling counterexamples to the widely accepted claim that non-restrictive relative clauses (NRCs) are always interpreted with respect to the global context, and never in the scope of entailment-canceling operators such as if. Local readings are available for NRCs attached to their host clause by a coordinating coherence relation. This paper develops a theoretical explanation of this pattern. We argue that NRCs are interpreted locally only if they are attached locally to their host clause both in syntax and in discourse structure. Subordinating coherence relations like Elaboration and Explanation resist discourse attachment in the scope of if because they tend to go together with relations that can only hold between speech acts. Like other subordinate clauses, NRCs tend to express subordinating coherence relations, which ultimately explains their pervasive tendency for global interpretation. In other words, this study shows how a theory of discourse coherence can help solve a problem in sentence semantics.