We give a description of the operad formed by the real locus of the moduli space of stable genus zero curves with marked points \overline{{\mathcal M}_{0,{n+1}}}({\mathbb R}) in terms of a homotopy quotient of an operad of associative algebras. We use this model to find different Hopf models of the algebraic operad of Chains and homologies of \overline{{\mathcal M}_{0,{n+1}}}({\mathbb R}). In particular, we show that the operad \overline{{\mathcal M}_{0,{n+1}}}({\mathbb R}) is not formal. The manifolds \overline{{\mathcal M}_{0,{n+1}}}({\mathbb R}) are known to be Eilenberg-MacLane spaces for the so called pure Cacti groups.
As an application of the operadic constructions we prove that for each n the cohomology ring H(\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R}),{\mathbb{Q}}) is a Koszul algebra and that the manifold \overline{{\mathcal M}_{0,{n+1}}}({\mathbb R}) is not formal but is a rational K(\pi,1) space. We give the description of the Lie algebras associated with the lower central series filtration of the pure Cacti groups.
