We compute the Poincaré polynomial and the cohomology algebra with rational coefficients of the manifold M n 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 M n . As was shown by the fourth author, the cohomology of M n 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 M n 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 L n of H .M n ; Q/ (associated to such quasibialgebras) factors through the the natural projection of L n to the associated graded Lie algebra of the prounipotent completion of the fundamental group of M n . 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 M n are not formal starting from n D 6.