$\mathcal{N}=3$ nonlinear multiplet and supergravity

We propose an $\mathcal{N}=3$ nonlinear multiplet coupled to conformal supergravity and use it to formulate the equations of motion for $\mathcal{N}=3$ Poincaré supergravity. These equations, which are naturally described in a new curved supergeometry with structure group $\mathsf{S}\mathsf{L}(2,\mathbb{C})$, imply that the $\mathcal{N}=3$ super-Bach tensor vanishes, and thus every solution of Poincaré supergravity is a solution of conformal supergravity. The aforementioned superspace formulation, which we refer to as $\mathcal{N}=3$ Einstein superspace, is described in terms of two dimension-$1\mathrm{/}2$ superfields: (i) the super-Weyl spinor $W_{\alpha }$; and (ii) a spinor isospinor ${\chi }_{\alpha }^i$.