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The first observation of the $B_s^0 \to \overline{D}^{*0} \phi$ decay isreported, with a significance of more than seven standard deviations, from ananalysis of $pp$ collision data corresponding to an integrated luminosity of 3fb$^{-1}$, collected with the LHCb detector at centre-of-mass energies of $7$and $8$ TeV. The branching fraction is measured relative to that of thetopologically similar decay $B^0 \to \overline{D}^0 \pi^+\pi^-$ and is found tobe $\mathcal{B}(B_s^0 \to \overline{D}^{*0} \phi) = (3.7 \pm 0.5 \pm 0.3 \pm0.2) \times 10^{-5}$, where the first uncertainty is statistical, the secondsystematic, and the third from the branching fraction of the $B^0 \to\overline{D}^0 \pi^+\pi^-$ decay. The fraction of longitudinal polarisation inthis decay is measured to be ${f_{\rm L} =(73 \pm 15 \pm 3)\%}$. The mostprecise determination of the branching fraction for the $B_s^0 \to\overline{D}^{0} \phi$ decay is also obtained, $\mathcal{B}(B_s^0 \to\overline{D}^{0} \phi) = (3.0 \pm 0.3 \pm 0.2 \pm 0.2) \times 10^{-5}$. Anupper limit, $\mathcal{B}(B^0 \to \overline{D}^{0} \phi) < 2.0 \ (2.2) \times10^{-6}$ 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.

Observation of $B_s^0 \to \overline{D}^{*0} \phi$ and search for $B^0 \to \overline{D}^0 \phi$ decays