Summary: A positive space is a space with a positive atlas, i.e., a collection of rational coordinate systems with subtraction free transition functions. The set of positive real points of a positive space is well defined. We define a tropical compactification of the latter. We show that it generalizes Thurston’s compactification of a Teichmüller space. A cluster Poisson variety, originally called cluster \({\mathcal X}\)-variety [the authors, Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865–930 (2009; Zbl 1180.53081)], is covered by a collection of coordinate tori \(({{\mathbb C}}^\ast)^n\), which form a positive atlas of a specific kind. We define a special completion \(\widehat{\mathcal X}\) of \({\mathcal X}\). It has a stratification whose strata are cluster Poisson varieties. The coordinate tori of \({\mathcal X}\) extend to coordinate affine spaces \({\mathbb A}^n\) in \(\widehat{\mathcal X}\). We define completions of Teichmüller spaces for decorated surfaces \({\mathbb S}\) with marked points at the boundary. The set of positive points of the special completion of the cluster Poisson variety \({\mathcal X}_{\mathrm{PGL} _2, {\mathbb S}}\) related to the Teichmüller theory on \({\mathbb S}\) [the authors, Publ. Math., Inst. Hautes Étud. Sci. 103, 1–211 (2006; Zbl 1099.14025)] is a part of the completion of the Teichmüller space (see Fig. 1 on the next page).