Flexible, algebraically unrealizable curves: Rehabilitation of Hilbert-Rohn-Gudkov approach

We give an isotopy classification of real pseudo-holomorphic and real algebraic M-curves of degree 8 on the quadratic cone arranged in some special way with respect to a line, and show that there exist real pseudo-holomorphic curves which are not isotopic to any real algebraic curve in this class. In a similar way we find a pseudoholomorphic real plane affine sextic which is not isotopic to a real algebraic sextic. The proofs are based on the braid group technique and highly singular degenerations of algebraic curves.