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.
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