  Zero-point quantum fluctuations of a N\'eel order can produce effective
interactions between quasi-orphan spins weakly coupled to the lattice. On the
$\sqrt{3}\times\sqrt{3}-$distorted triangular lattice, this phenomenon leads to
a correlated partially disordered phase. In this article, we use matrix product
state methods to study a similar model: the $S=1/2$ stuffed square lattice.
Tunning the exchange amplitudes we go from a square lattice plus orphan central
spins at $J'/J =0$, to the union jack lattice at $J'/J=1$, and a square lattice
including all spins at $J/J'=0$. We calculate the complete antiferromagnetic
phase diagram, dominated by ferrimagnetic and N\'eel orders, and compare with
existing results. Most importantly, we find a partially disordered phase in the
weakly frustrated regime. In this phase, the N\'eel order from the square
lattice is unaffected, while the central spins form a collective state with
exponentially decaying double striped correlations. We also study the role of
quantum fluctuations by introducing an ordering staggered magnetic field on the
square sublattice, and find that the central spins order ferromagnetically when
fluctuations from the N\'eel order are suppressed.
