AI agents can each make locally valid decisions yet jointly produce an invalid result. We call this the global coherence problem: a failure of shared state, not merely of intelligence. Our Observation-Aliasing Impossibility Theorem gives the exact boundary. A policy can guarantee a valid action exactly when all worlds producing the same observation share an admissible action. If k indistinguishable worlds require pairwise-disjoint actions, the best randomized worst-case success is 1/k; more reasoning, roles, messages, or samples cannot recover the missing distinction. A stronger model can reason better within its context, but it cannot see beyond it. The smallest sufficient context is NP-hard to find, yet counterexample-guided refinement finds a sufficient one and recertifies it as rules change. We give local-to-global runtime semantics X = (H, C, G, F; D): topology H records overlapping scopes; category C governs state-changing actions; groupoid G keeps reversible translations; sheaf F tests whether local views glue into one world; and history D keeps only what alters legal futures. Each preserves the needed information as state is split, translated, or changed. Models propose; the harness owns shared state and governs commit. Ten studies test the boundary. On a revision benchmark, a frontier model scores 40/40 when the deciding event is visible; hidden, every arm is consistent with chance (1/3); one restored fact returns 40/40. On TeamBench, ordinary teams exceed a shared budget in 5/5 runs, a visible live count leaves 4/5, and commit enforcement leaves 0/5. In tau2-bench Telecom, current-state checks score 0.07 after silent reverts; the harness scores 1.00. Refinement keeps a sufficient context through rule changes at 28% of the all-field cost. Solvers that already own the state tie it, as predicted. Local intelligence cannot substitute for missing global state.

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