We complete a fiberwise isotopy classification of smooth real algebraic and pseudoholomorphic curves of degree 8 on the quadratic cone, which have a specially shaped oval crossing a given generating line of the cone in four real points. We link this classification with an isotopy classification of smoothing of real plane curve singularity which is the union of four smooth real local branches quadratically tangent to each other. (the singularity $X_{21}$).