We study the question of whether the quasilocality (continuity, almost-Markovianness) property of Gibbs measures remains valid under a projection on a sub-σ-algebra. Our method is based on the construction of global specifications, whose projections yield local specifications for the projected measures. For Gibbs measures compatible with monotonicity-preserving local specifications, we show that the set of configurations where quasilocality is lost is an event of the tail field. This set is

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... n to be empty whenever a strong-uniqueness property is satisfied, and of measure zero when the original specification admits a single Gibbs measure. Moreover, we provide a criterion for non-quasilocality (based on a quantity related to the surface-tension). We apply these results to projections of the extremal measures of the Ising model. In particular, our non-quasilocality criterion allows us to extend and make more complete previous studies of projections to a sublattice of one less dimension (Schonmann example).