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We study the full counting statistics (FCS) of quantum gases in samples of thousands of interacting bosons, detected atom by atom after a long free-fall expansion. In this far-field configuration, the FCS reveals the many-body coherence from which we characterize iconic states of interacting lattice bosons by deducing their normalized correlations g^{(n)}(0) up to the order n=6. In Mott insulators, we find a thermal FCS characterized by perfectly contrasted correlations g^{(n)}(0)=n!. In interacting Bose superfluids, we observe small deviations to the Poisson FCS and to the ideal values g^{(n)}(0)=1 expected for a pure condensate. These deviations become larger as we increase the interaction strength and reveal the role of the quantum depletion of the condensate on many-body coherence. The approach to many-body correlations demonstrated here is readily extendable to characterize a large variety of interacting quantum states and phase transitions.