  Using the parton construction, we build a three-dimensional (3D) multilayer
fractional quantum Hall state with average filling \nu = 1/3 per layer that is
qualitatively distinct from a stacking of weakly coupled Laughlin states. The
state supports gapped charge e/3 fermionic quasiparticles that can propagate
both within and between the layers, in contrast to the quasiparticles in a
multilayer Laughlin state which are confined within each layer. Moreover, the
state has gapless neutral collective modes, a manifestation of an emergent
"photon", which is minimally coupled to the fermionic quasiparticles. The
surface sheath of the multilayer state resembles a chiral analog of the
Halperin-Lee-Read state, which is protected against gap forming instabilities
by the topological character of the bulk 3D phase. We propose that this state
might be present in multilayer systems in the "intermediate tunneling regime",
where the interlayer tunneling strength is on the same order as the Coulomb
energy scale. We also find that the parton construction leads to a candidate
state for a bilayer \nu = 1/3 system in the intermediate tunneling regime. The
candidate state is distinct from both a bilayer of \nu=1/3 Laughlin states and
the single layer \nu = 2/3 state, but is nonetheless a fully gapped fractional
quantum Hall state with charge e/3 anyonic quasiparticles.
