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The supertwistor and bi-supertwistor formulations for ${\mathcal N}$-extendedanti-de Sitter (AdS) superspace in four dimensions, ${\rm AdS}^{4|4\mathcalN}$, were derived two years ago in arXiv:2108.03907. In the present paper, weintroduce a novel realisation of the ${\mathcal N}$-extended AdS supergroup$\mathsf{OSp}(\mathcal{N}|4;\mathbb{R})$ and apply it to develop a cosetconstruction for ${\rm AdS}^{4|4\mathcal N}$ and the corresponding differentialgeometry. This realisation naturally leads to an atlas on ${\rmAdS}^{4|4\mathcal N}$ (that is a generalisation of the stereographic projectionfor a sphere) that consists of two charts with chiral transition functions for${\mathcal N}>0$. A manifestly $\mathsf{OSp}(\mathcal{N}|4;\mathbb{R})$invariant model for a superparticle in ${\rm AdS}^{4|4\mathcal N}$ is proposed.Additionally, by employing a conformal superspace approach, we describe themost general conformally flat $\mathcal N$-extended supergeometry. Thisconstruction is then specialised to the case of ${\rm AdS}^{4|4\mathcal N}$.

Embedding formalism for ${\mathcal N}$-extended AdS superspace in four dimensions