  We study when a continuous isometric action of a Polish group on a complete
metric space is, or can be, transitive. Our main results consist of showing
that certain Polish groups, namely $\mathrm{Aut}^*(\mu)$ and
$\mathrm{Homeo}^+[0,1]$, such an action can never be transitive (unless the
space acted upon is a singleton). We also point out "circumstantial evidence"
that this pathology could be related to that of Polish groups which are not
closed permutation groups and yet have discrete uniform distance, and give a
general characterisation of continuous isometric action of a Roeckle-precompact
Polish group on a complete metric space is transitive. It follows that the
morphism from a Roeckle-precompact Polish group to its Bohr compactification is
surjective.
