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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 π-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 “Néel order” features a nonlocal string order parameter for an in-plane Néel component, in addition to an anisotropic local order parameter. The associated quantum order is characterized by an emergent Z_{2}×Z_{2} gauge field when the Néel vector is along the z[over ̂] direction, and a 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 Z_{2}×Z_{2} phase is fractionalized and nonlocal.