The aim of the paper is to attach a noncommutative cluster-like structure to each marked surface $Sigma$. This is a noncommutative algebra ${mathcal A}_Sigma$ generated by noncommutative geodesics between marked points subject to certain triangle relations and noncommutative analogues of Ptolemy-Plucker relations. It turns out that the algebra ${mathcal A}_Sigma$ exhibits a noncommutative Laurent Phenomenon with respect to any triangulation of $Sigma$, which confirms its cluster nature. As a surprising byproduct, we obtain a new topological invariant of $Sigma$, which is a free or a 1-relator group easily computable in terms of any triangulation of $Sigma$. Another application is the proof of Laurentness and positivity of certain discrete noncommutative integrable systems.
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