Multi-component LLM agents assemble probabilistic claims from components that each see only part of a joint problem; the composition can violate basic probability axioms even when every component is locally coherent. We formalise this locally coherent, globally incoherent failure via the compositional residual eps*, the L2 distance from the composed quote to the joint coherent polytope, computable at runtime from system output and the declared cross-component coupling constraints. A product-structure dichotomy characterises when local coherence suffices, and a Rayleigh-quotient prediction matches the observed residual within 7% on three of four relation classes. A hierarchical Boyle-Dykstra projection repairs the composition deterministically; an anytime-valid e-process gives sequential coherence monitoring. Across 1,876 ensemble cliques on a four-LLM mid-tier panel (frontier-panel rerun in Section 5.5), eps* > 0 on 33-94% of cliques, translating to +0.115 nats per bet of regret on 1,770 resolved bets under the proportional allocation rule (the gain collapses to +0.006 under bettors that themselves coherentise). Three intuitive LLM-side mitigations(retrieval, partition-aware prompting, aggregator-LLM) each fail or regress.