With any non-necessarily orientable unpunctured marked surface (S,M)\documentclass[12pt]{minimal}
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\begin{document}$$({\mathbf {S}},{\mathbf {M}})$$\end{document}, we associate a commutative algebra A(S,M)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}_{({\mathbf {S}},{\mathbf {M}})}$$\end{document}, called quasi-cluster algebra, equipped with a distinguished set of generators, called quasi-cluster variables, in bijection with the set of arcs and one-sided simple closed curves in (S,M)\documentclass[12pt]{minimal}
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\begin{document}$$({\mathbf {S}},{\mathbf {M}})$$\end{document}. Quasi-cluster variables are naturally gathered into possibly overlapping sets of fixed cardinality, called quasi-clusters, corresponding to maximal non-intersecting families of arcs and one-sided simple closed curves in (S,M)\documentclass[12pt]{minimal}
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\begin{document}$$({\mathbf {S}},{\mathbf {M}})$$\end{document}. If the surface S\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {S}}$$\end{document} is orientable, then A(S,M)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}_{({\mathbf {S}},{\mathbf {M}})}$$\end{document} is the cluster algebra associated with the marked surface (S,M)\documentclass[12pt]{minimal}
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\begin{document}$$({\mathbf {S}},{\mathbf {M}})$$\end{document} in the sense of Fomin, Shapiro and Thurston. We classify quasi-cluster algebras with finitely many quasi-cluster variables and prove that for these quasi-cluster algebras, quasi-cluster monomials form a linear basis. Finally, we attach to (S,M)\documentclass[12pt]{minimal}
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\begin{document}$$({\mathbf {S}},{\mathbf {M}})$$\end{document} a family of discrete integrable systems satisfied by quasi-cluster variables associated to arcs in A(S,M)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}_{({\mathbf {S}},{\mathbf {M}})}$$\end{document} and we prove that solutions of these systems can be expressed in terms of cluster variables of type A\documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document}.
