Almost Einstein manifolds satisfy a generalisation of the Einstein condition; they are Einstein on an open dense subspace and, in general, have a conformal scale singularity set $\Sigma$ that is a conformal infinity for the Einstein metric. In case an almost Einstein manifold is closed and $\Sigma$ is a hypersurface we call the corresponding Einstein metric conformally closed. Such Einstein metrics represent a subclass of conformally compact Poincaré-Einstein metrics. With respect to a special defining function of the boundary every Poincaré-Einstein metric can be expressed in normal form. Similarly, in this paper we discuss closed manifolds, which admit multiple almost Einstein structures, whose scale singularity sets intersect non-trivially. In a neighbourhood of that intersection set $\Sigma(\cal S)$ we describe the underlying conformal geometry by normal form metrics and the $S^l$-doubling construction of [24]. The set $\Sigma(\cal S)$ is a totally umbilic submanifold (of higher codimension).