  All previous cuprate superconductors display a set of common features: (i)
vicinity to a Cu 3$d^{9}$ configuration; (ii) separated CuO$_2$ planes; (iii)
superconductivity for doping $\delta \sim$ 0.1$-$0.3. Recently [PNAS {\bf 24},
12156 (2019)] challenged this picture by discovering "highly overdoped"
superconducting Ba$_2$CuO$_{3+y}$. Using density-functional theory + dynamical
mean-field theory, we reveal a bilayer structure of Ba$_2$CuO$_{3.2}$ of
alternating quasi 2D and quasi 1D character. Correlations tune an inter-layer
self-doping leading to an almost half-filled, strongly nested quasi 1D
$d_{b^2-c^2}$ band, which is prone to strong antiferromagnetic fluctuations,
possibly at the origin of superconductivity in Ba$_2$CuO$_{3+y}$.
