The aim of the paper is to define noncommutative cluster structure on several algebras ${ \mathcal A}$ related to marked surfaces possibly with orbifold points of various orders, which includes noncommutative clusters, i.e., embeddings of a given group $G$ into the multiplicative monoid ${ \mathcal A}^\times$ and an action of a certain braid-like group $Br_{ \mathcal A}$ by automorphisms of each cluster group in a compatible way. For punctured surfaces we construct new symmetries, noncommutative tagged clusters and establish a noncommutative Laurent Phenomenon.
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