AI agents share code, documents, tools, and analyses. No participant sees the whole state, so every step can look right while the combined result is wrong. We call this the global coherence problem. It is a problem of state, not only of intelligence: a stronger model decides better locally, but cannot see what is missing from its context. The Observation-Aliasing Impossibility Theorem gives the exact condition. An agent can always act validly exactly when every world consistent with its context allows a common action. When look-alike worlds need different actions, no reasoning, role, message, or sample can guarantee the right one; with k such worlds whose valid actions are disjoint, the best randomized policy succeeds with probability 1/k. Finding the smallest sufficient context is NP-hard, but refinement finds one and keeps it current as rules change. A second failure survives full information: valid pieces can fail to fit into one world. The same theorem sets the conditions for coherence; sheaves and groupoids are one way to meet them. Models propose, and the harness decides what is valid. Ten studies test it. On TeamBench, teams overspend a shared budget in 5/5 runs; showing them the live count still leaves 4/5, and a harness that owns the budget leaves 0/5. Identity collisions that no pairwise check can see appear in 23 of 45 public alignment networks; split-view agents create them in 9 of 40 merges, and a network check removes all nine. A frontier LLM solves 40/40 revision tasks when it sees the deciding event; with the event hidden, no arm that reasons longer, votes, or splits into roles significantly beats chance, and a companion study replicates this on models from a second provider. In tau2-bench Telecom, after a silent revert, the harness scores 1.00 against 0.07 for call-time policy checks. Refinement keeps a sufficient context at 28% of the cost of sharing every field.

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