  Let $G$ be a Polish group and let $H \leq G$ be a compact subgroup. We prove
that there exists a Borel set $T \subset G$ which is simultaneously a complete
set of coset representatives of left and right cosets, provided that a certain
index condition is satisfied. Moreover, we prove that this index condition
holds provided that $G$ is locally compact and $G/G^\circ$ is compact or $H$ is
a compact Lie group. This generalizes a result which is known for discrete
groups under various finiteness assumptions, but is known to fail for general
inclusions of infinite groups. As an application, we prove that Bohr closed
subgroups of countable, discrete groups admit common transversals.
