We compute the Poincaré polynomial and the cohomology algebra with rational coefficients of the manifold Mn of real points of the moduli space of algebraic curves of genus 0 with n labeled points. This cohomology is a quadratic algebra, and we conjecture that it is Koszul. We also compute the 2-local torsion in the cohomology of Mn. As was shown by the fourth author, the cohomology of Mn does not have odd torsion, so that the above determines the additive structure of the integral homology and cohomology. Further, we prove that the rational homology operad of Mn is the operad of 2-Gerstenhaber algebras, which is closely related to the Hanlon-Wachs operad of 2-Lie algebras (generated by a ternary bracket). Finally, using Drinfeld’s theory of quantization of coboundary Lie quasibialgebras, we show that a large series of representations of the quadratic dual Lie algebra Ln of H∗(Mn,Q) (associated to such quasibialgebras) factors through the the natural projection of Ln to the associated graded Lie algebra of the prounipotent completion of the fundamental group of Mn. This leads us to conjecture that the said projection is an isomorphism, which would imply a formula for lower central series ranks of the fundamental group. On the other hand, we show that the spaces Mn are not formal starting from n=6.