  We carry out a sign-problem-free quantum Monte Carlo calculation of a bilayer
model with a repulsive intra-layer Hubbard interaction and a ferromagnetic
inter-layer interaction. The latter breaks the global $SU(2)$ spin rotational
symmetry but preserves a $U(2)\times U(2)$ invariance under mixing of same-spin
electrons between layers. We show that despite the difference in symmetry, the
bilayer model exhibits the same qualitative features found in the single-layer
Hubbard model. These include stripe phases, whose nature is sensitive to the
presence of next-nearest-neighbor hopping, a maximum in the Knight shift that
moves to lower temperatures with increasing hole doping, and lack of evidence
for intra-layer d-wave superconductivity. Instead, we find a superconducting
phase, coexisting with stripes, whose critical temperature traces a dome as a
function of doping and is due to inter-layer spin-polarized pairing that is
induced by the ferromagnetic interaction.
