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We give a description of the operad formed by the real locus of the modulispace of stable genus zero curves with marked points $\overline{{\mathcalM}_{0,{n+1}}}({\mathbb R})$ in terms of a homotopy quotient of an operad ofassociative algebras. We use this model to find different Hopf models of thealgebraic operad of Chains and homologies of $\overline{{\mathcalM}_{0,{n+1}}}({\mathbb R})$. 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 beEilenberg-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 $H(\overline{{\mathcal M}_{0,{n+1}}}({\mathbbR}),{\mathbb{Q}})$ is a Koszul algebra and that the manifold$\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal but is arational $K(\pi,1)$ space. We give the description of the Lie algebrasassociated with the lower central series filtration of the pure Cacti groups.

Real moduli space of stable rational curves revisted