Multi-agent debate commonly exchanges complete rationales even when disagreements concern only a few intermediate claims. We introduce Localized Multi-Agent Debate (LMAD), an inference-time protocol that represents agent rationales as nodes, locates their earliest conflict, and restricts debate to the corresponding local segments. Guarded resolution extends a shared committed state so that later conflicts can be addressed without reopening accepted steps. We evaluate LMAD on four multi-hop question-answering benchmarks using ten backbones from four model families. Our method achieves the highest macro-averaged judge accuracy across all ten backbones, outperforming the strongest conventional baseline by up to 7.20 percentage points.