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The problem of separate zero-error coding of correlated sources is considered. Inner and outer single-letter bounds are established for the achievable rate region, and conditions for their coincidence are investigated. It is shown that successive encoding combined with time sharing is not always an optimal coding strategy. Conditions for its optimality are derived. The inner bound to the achievable rate region follows as a special case of the single-letter characterization of a generalized zero-error multiterminal rate-distortion problem. The applications of this characterization to a problem of remote computing are also explored. Other results include (i) a product-space characterization of the achievable rates, (ii) bounds for finite block length, and (iii) asymptotic fixed-length rates.