  We construct a family of short-range resonating-valence-bond wave functions
on a layered cubic lattice, allowing for a tunable anisotropy in the amplitudes
assigned to nearest-neighbour valence bonds along one axis. Monte Carlo
simulations reveal that four phases are stabilized over the full range of the
anisotropy parameter. They are separated from one another by a sequence of
continuous quantum phase transitions. An antiferromagnetic phase, centred on
the perfect isotropy point, intervenes between two distinct quantum spin liquid
states. One of them is continuously deformable to the two-dimensional U(1) spin
liquid, which is known to exhibit critical bond correlations. The other has
both spin and bond correlations that decay exponentially. The existence of this
second phase is proof that, contrary to expectations, neither a bipartite
lattice structure nor a conventional Marshall sign rule is an impediment to
realizing a fully gapped quantum spin liquid.
