  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.
