  We present the quantitative phase diagram of the bilayer bosonic fractional
quantum Hall system on the torus geometry at total filling factor $\nu=1$ in
the lowest Landau level. We consider short-range interactions within and
between the two layers, as well as the inter-layer tunneling. In the fully
polarized regime, we provide an updated detailed numerical analysis to
establish the presence of the Moore-Read phase of both even and odd numbers of
particles. In the actual bilayer situation, we find that both inter-layer
interactions and tunneling can provide the physical mechanism necessary for the
low-energy physics to be driven by the fully polarized regime, thus leading to
the emergence of the Moore-Read phase. Inter-layer interactions favor a
ferromagnetic phase when the system is $SU(2)$ symmetric, while the inter-layer
tunneling acts as a Zeeman field polarizing the system. Besides the Moore-Read
phase, the $(220)$ Halperin state and the coupled Moore-Read state are also
realized in this model. We study their stability against each other.
