  The first observation of the $B_s^0 \to \overline{D}^{*0} \phi$ decay is
reported, with a significance of more than seven standard deviations, from an
analysis of $pp$ collision data corresponding to an integrated luminosity of 3
fb$^{-1}$, collected with the LHCb detector at centre-of-mass energies of $7$
and $8$ TeV. The branching fraction is measured relative to that of the
topologically similar decay $B^0 \to \overline{D}^0 \pi^+\pi^-$ and is found to
be $\mathcal{B}(B_s^0 \to \overline{D}^{*0} \phi) = (3.7 \pm 0.5 \pm 0.3 \pm
0.2) \times 10^{-5}$, where the first uncertainty is statistical, the second
systematic, and the third from the branching fraction of the $B^0 \to
\overline{D}^0 \pi^+\pi^-$ decay. The fraction of longitudinal polarisation in
this decay is measured to be ${f_{\rm L} =(73 \pm 15 \pm 3)\%}$. The most
precise determination of the branching fraction for the $B_s^0 \to
\overline{D}^{0} \phi$ decay is also obtained, $\mathcal{B}(B_s^0 \to
\overline{D}^{0} \phi) = (3.0 \pm 0.3 \pm 0.2 \pm 0.2) \times 10^{-5}$. An
upper limit, $\mathcal{B}(B^0 \to \overline{D}^{0} \phi) < 2.0 \ (2.2) \times
10^{-6}$ at $90\%$ (95\%) confidence level is set. A constraint on the
$\omega-\phi$ mixing angle $\delta$ is set at $|\delta| < 5.2^\circ~
(5.5^\circ)$ at $90\%$ ($95\%$) confidence level.
