  We have followed up two ultra-diffuse galaxies (UDGs), detected adjacent to
stellar streams, with Hubble Space Telescope (HST) imaging and HI mapping with
the Jansky Very Large Array (VLA) in order to investigate the possibility that
they might have a tidal origin. With the HST F814W and F555W images we measure
the globular cluster (GC) counts for NGC 2708-Dw1 and NGC 5631-Dw1 as
$2^{+1}_{-1}$ and $5^{+1}_{-2}$, respectively. NGC 2708-Dw1 is undetected in HI
down to a 3$\sigma$ limit of $\log (M_\mathrm{HI}/\mathrm{M_\odot}) = 7.3$, and
there is no apparent HI associated with the nearby stellar stream. There is a
2$\sigma$ HI feature coincident with NGC 5631-Dw1. However, this emission is
blended with a large gaseous tail emanating from NGC 5631 and is not
necessarily associated with the UDG. The presence of any GCs and the lack of
clear HI connections between the UDGs and their parent galaxies strongly
disfavor a tidal dwarf galaxy origin, but cannot entirely rule it out. The GC
counts are consistent with those of normal dwarf galaxies, and the most
probable formation mechanism is one where these UDGs were born as normal dwarfs
and were later tidally stripped and heated. We also identify an over-luminous
($M_\mathrm{V} = -11.1$) GC candidate in NGC 2708-Dw1, which may be a nuclear
star cluster transitioning to an ultra-compact dwarf as the surrounding dwarf
galaxy gets stripped of stars.
