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.
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