Tensionless strings and the space-time signature change

We construct a tensionless string model in a four-dimensional space-time ${\bf R}^{2,2}$ with the (2,2) signature and solve the string equations. We find that the signature change radically changes the structure of the holonomy group of the null worldsheet. As a result the introduction of the worldsheet Dirac operator invariant under the holonomy group becomes possible. We show that the such possibility is absent in the tensionless string model in the (3+1)-dimensional Minkowski space.