  A real algebraic link in the 3-sphere is defined as the zero locus in the
3-sphere of a real algebraic function from $\mathbb{R}^4$ to $\mathbb{R}^2$. A
real algebraic open book decomposition on the 3-sphere is by definition the
Milnor fibration of such a real algebraic function, in case it exists. We prove
that every overtwisted contact structure on the 3-sphere with positive three
dimensonal invariant $d_3$ (apart from possibly 9 exceptions) are real
algebraic. Our construction shows in particular that the pages of the
associated open books are planar.
