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 X is covered by a collection of coordinate tori , whichform a positive atlas of a specific kind. We define a special completion $\widehat{\mathcal{X}}$ of ${\mathcal{X}$. It has astratification whose strata are cluster Poisson varieties. The coordinate tori of X extend tocoordinate affine spaces $\mathbb{A}^n$ in $\widehat{\mathcal{X}}$. We define completions of Teichmüller spaces for decorated surfaces $\mathbb{S}$ with marked pointsat the boundary. The set of positive points of the special completion of the cluster Poissonvariety $\mathcal{X}_{PGL(2,\mathbb S) related to the Teichmüller theory on $\mathbb{S}$ [FG1] is a part of the completion ofthe Teichmüller space.