The second-order achievable rate region in Slepian-Wolf source coding systems is investigated. The concept of second-order achievable rates, which enables us to make a finer evaluation of achievable rates, has already been introduced and analyzed for general sources in the single-user source coding problem. Accordingly, in this paper, we first define the second-order achievable rate region for the Slepian-Wolf coding system and establish the source coding theorem for general sources in the second-order sense. Moreover, we compute the explicit second-order achievable rate region for i.i.d. correlated sources with countably infinite alphabets and mixed correlated sources, respectively, using the relevant asymptotic normality.