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A non-algebraic patchwork

Itenberg and Shustin's pseudoholomorphic curve patchworking is in principle more flexible than Viro's original algebraic one. It was natural to wonder if the former method allows one to construct non-algebraic objects. In this paper we construct the first examples of patchworked real pseudoholomorphic curves in rational geometrically ruled surfaces whose position with respect to the pencil of lines cannot be realised by any homologous real algebraic curve.