  We study the ground-state properties of a spin-1 Heisenberg model on a square
lattice with the first- and second-nearest-neighbor antiferromagnetic couplings
$J_1$ and $J_2$ and a three-spin scalar chirality term $J_\chi$. Using the
density matrix renormalization group calculation, we map out a global phase
diagram including various magnetic order phases and an emergent quantum spin
liquid phase. The nature of the spin liquid is identified as a bosonic
non-Abelian Moore-Read state from the fingerprint of the entanglement spectra
and identification of a full set of topological sectors. We further unveil a
stripe magnetic order coexisting with this spin liquid. Our results not only
establish a rare example of non-Abelian spin liquids in simple spin systems but
also demonstrate the coexistence of fractionalized excitations and magnetic
order beyond mean-field descriptions.
