  We deliver examples of non-Gromov hyperbolic tube domains with convex bases
(equipped with the Kobayashi distance). This is shown by providing a criterion
on non-Gromov hyperbolicity of (non-smooth) domains.The results show the
similarity of geometry of the bases of non-Gromov hyperbolic tube domains with
the geometry of non-Gromov hyperbolic convex domains. A connection between the
Hilbert metric of a convex domain $\Omega$ in $\mathbb R^n$ with the Kobayashi
distance of the tube domain over the domain $\Omega$ is also shown. Moreover,
continuity properties up to the boundary of complex geodesics in tube domains
with a smooth convex bounded base are also studied in detail.
