  Motivated by a recent experiment [Revelle et al. Phys. Rev. Lett. 117, 235301
(2016)] that characterized the one- to three-dimensional crossover in a
spin-imbalanced ultracold gas of $^6$Li atoms trapped in a two-dimensional
array of tunnel-coupled tubes, we calculate the phase diagram for this system
using Hartree-Fock Bogoliubov-de Gennes mean-field theory, and compare the
results with experimental data. Mean-field theory predicts fully spin-polarized
normal, partially spin-polarized normal, spin-polarized superfluid, and
spin-balanced superfluid phases in a homogeneous system. We use the local
density approximation to obtain density profiles of the gas in a harmonic trap.
We compare these calculations with experimental measurements in Revelle {\em et
al.} as well as previously unpublished data. Our calculations qualitatively
agree with experimentally-measured densities and coordinates of the phase
boundaries in the trap, and quantitatively agree with experimental measurements
at moderate-to-large polarizations. Our calculations also reproduce the
experimentally-observed universal scaling of the phase boundaries for different
scattering lengths at a fixed value of scaled inter-tube tunneling. However,
our calculations have quantitative differences with experimental measurements
at low polarization, and fail to capture important features of the one- to
three-dimensional crossover observed in experiments. These suggest the
important role of physics beyond-mean-field theory in the experiments. We
expect that our numerical results will aid future experiments in narrowing the
search for the FFLO phase.
