  We study solutions of type IIB string theory dual to ${\cal N}=4$
supersymmetric Yang-Mills theory on half of $\mathbb{R}^{3,1}$ coupled to
holographic three-dimensional superconformal field theories (SCFTs) at the edge
of this half-space. The dual geometries are asymptotically $AdS^5 \times S^5$
with boundary geometry $\mathbb{R}^{2,1} \times \mathbb{R}^+$, with a
geometrical end-of-the-world (ETW) brane cutting off the other half of the
asymptotic region of the would-be Poincar\'e $AdS^5 \times S^5$. We show that
by choosing the 3D SCFT appropriately, this ETW brane can be pushed arbitrarily
far towards the missing asymptotic region, recovering the "missing" half of
Poincar\'e $AdS^5 \times S^5$. We also show that there are 3D SCFTs whose dual
includes a wedge of Poincar\'e $AdS^5 \times S^5$ with an angle arbitrarily
close to $\pi$, with geometrical ETW branes on either side.
