Algebraically unrealizable complex orientations of plane real pseudoholomorphic curves

We prove two inequalities for the complex orientations of a separating non-singular real algebraic curve in $${\mathbb {RP}}^2$$ of any odd degree. We also construct a separating non-singular real (i.e., invariant under the complex conjugation) pseudoholomorphic curve in $${\mathbb {CP}}^2$$ of any degree congruent to 9 mod 12 which does not satisfy one of these inequalities. Therefore the oriented isotopy type of the real locus of each of these curves is algebraically unrealizable.