  The quantum BRST charges for the Bershadsky-Knizhnik orthogonal
quasi-superconformal algebras are constructed. These two-dimensional
superalgebras have the $N$-extended non-linearly realised supersymmetry and the
$SO(N)$ internal symmetry. The BRST charge nilpotency conditions are shown to
have a unique solution at $N>2$, namely, $N=4$ and $k=-2$, where $k$ is central
extension parameter of the Ka\v{c}-Moody subalgebra. We argue about the
existence of a new string theory with the non-linearly realised $N=4$
world-sheet supersymmetry and negative `critical dimension'.
