  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
$\Sigma_n$ whose position with respect to the pencil of lines cannot be
realised by any homologous real algebraic curve.
