  We study the existence problem for tilted unduloids in
$\mathbb{H}^2\times\mathbb{R}$. These are singly periodic annuli with constant
mean curvature $H>1/2$ in $\mathbb{H}^2\times\mathbb{R}$, and the periodicity
of these surfaces is with respect to a discrete group of translations along a
geodesic that is neither vertical nor horizontal in the Riemannian product
$\mathbb{H}^2\times\mathbb{R}$. Via the Daniel correspondence we are able to
reduce this existence problem to a uniqueness problem in the Berger spheres: if
a pair of linked horizontal geodesics bounds exactly two embedded minimal
annuli (for a fixed orientation of the boundary curves) then tilted unduloids
in $\mathbb{H}^2\times\mathbb{R}$ exist.
