  We study the near-threshold incoherent $\phi$ photoproduction on the deuteron
based on a model of $\gamma N \to \phi N$, consisting of Pomeron, $(\pi, \eta)$
exchanges, and a $J^P = 3/2^-$ resonance, which describes the low energy
$\gamma p \to \phi p$ LEPS data well, including the peak in the forward
differential cross section.
  The calculation is done up to double rescatterings, with the spin dependence
of the elementary $\gamma N \to \phi N$ amplitude retained throughout the
calculation. The Fermi motion and final-state interactions (FSI) are all
properly treated as prescribed by realistic nucleon-nucleon interaction. The
couplings of the resonance to $\gamma n$ and $\phi n$ channels are estimated
with the help of a constituent quark model. The main features of the LEPS and
CLAS data are described reasonably well except for some quantitative
discrepancies at very low energies and low momentum transfers regions. It is
found that contributions of Fermi motion, $pn$ FSI, and resonance are all
indispensable in bridging the differences between the single-scattering results
and the data. The off-shell rescattering is found to be important as it cancels
out a large portion of the on-shell contribution. The discrepancies at low
momentum transfer region might be related to the binning size of the data. No
peak is found to be associated with the weak resonance as it gets smeared out
by the Fermi motion and FSI with deuterium target. The problem at very low
energy region hints at the possible contributions from other mechanisms and
should be investigated in depth with the use of recent high statistics $\gamma
p \to \phi p$ data from CLAS.
