A search for the decays of the Higgs and $Z$ bosons to a $\ensuremath{\phi}$ meson and a photon is performed with a $pp$ collision data sample corresponding to an integrated luminosity of $2.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ collected at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ with the ATLAS detector at the LHC. No significant excess of events is observed above the background, and 95% confidence level upper limits on the branching fractions of the Higgs and $Z$ boson decays to $\ensuremath{\phi}\ensuremath{\gamma}$ of $1.4\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}3}$ and $8.3\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}6}$, respectively, are obtained.