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A new class of twistor-like string models in four-dimensional space-time extended by the addition of six tensorial central charge (TCC) coordinates z mn is studied. The hamiltonian of tensionless string in the extended space-time is derived and its symmetries are investigated. We establish that the string constraints reduce the number of independent TCC coordinates z mn to one real effective coordinate which composes an effective 5-dimensional target space together with the x m coordinates. We construct the P.B. algebra of the first class constraints and discover that it coincides with the P.B. algebra of tensionless strings. The Lorentz covariant antisymmetric DiracĈ-matrix of the P.B. of the second class constraints is constructed and its algebraic structure is further presented.