It goes back to Ahlfors that a real algebraic curve C admits a separating morphism f to the complex projective line if and only if the real part of the curve disconnects its complex part, i.e. the curve is separating. The degree of such morphism f is bounded from below by the number l of real connected components of the real locus of C. The sharpness of this bound is not a priori clear. We prove that real algebraic separating curves, embedded in some ambient surface and with l bounded in a certain way, do not admit separating morphisms of lowest possible degree. Moreover, this result of non-existence can be applied to show that certain real separating plane curves of degree d, do not admit totally real pencils of curves of degree k for k<l/d +1.