Same Answer, Different Reasons: Disentangling Coordination from Consensus in Multi-Agent Debate

Large language model agents in a multi-agent debate can reach the same final answer for two different reasons: because they saw the same evidence, or because a coalition among them privately coordinated to produce it. From the verdict alone the two cases are indistinguishable, letting a coordinated minority pass as independent consensus. We cast this as an identifiability problem: recovering a latent partition of agents into coalition and non-coalition members from public information alone (transcript and shared task dossier). To address it, we introduce **LATTICE**, a Layered Agreement Tracing framework that grounds each message into atomic claims, judges typed, directed relations between claims of different agents, and aggregates them into time-windowed, multi-lens agent-by-agent matrices under a composite with pre-specified weights. A stance-correlation baseline is kept out of the composite, and every result is checked against a paired negative-control session with the same dossier and roster but no coalition. Under this protocol, the composite separates the latent agent partition more reliably than stance correlation, does not recover the transferred partition in the negative controls, achieves the strongest pairwise ranking among public-information detectors, and, when the coalition size is known, recovers the installed coalition exactly, matching group-structure detectors without their high false-structure rates. On external benchmarks it keeps its ranking advantage where stance correlation falls to chance, recovers coalitions well above chance but not exactly, and does not mistake honest, information-driven convergence for coordination. Code and data are available [here](https://anonymous.4open.science/r/lattice-500A).