  We study the phase diagram of a bilayer quantum spin liquid model with
Kitaev-type interactions on a square lattice. We show that the low energy limit
is described by a $\pi$-flux Hubbard model with an enhanced SO(4) symmetry. The
antiferromagnetic Mott transition of the Hubbard model signals a magnetic
fragmentation transition for the spin and orbital degrees of freedom of the
bilayer. The fragmented "N\'eel order" features a non-local string order
parameter for an in-plane N\'eel component, in addition to an anisotropic local
order parameter. The associated quantum order is characterized by an emergent
$\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ gauge field when the N\'eel vector is
along the $\hat{z}$ direction, and a $\mathbb{Z}_2$ gauge field otherwise. We
underpin these results with a perturbative calculation, which is consistent
with the field theory analysis. We conclude with a discussion on the low energy
collective excitations of these phases and show that the Goldstone boson of the
$\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ phase is fractionalized and non-local.
