I’m fairly certain there isn’t an English common name for this species. While researching this species yesterday due to some confusion over its scientific name and description, we (Douglas, Eric and myself) learned that little work had been done on the taxonomy of the genus Ephedra since the late 19th century. Some modern work has occurred in the past fifteen years or so, but it certainly hasn’t trickled into the horticultural literature yet. Of the older horticultural texts we examined, it seemed like the descriptions of Ephedra were all slight variations from the late 19th century work. Ephedra has often been regarded as having little ornamental value, though perhaps that will no longer be true with changing tastes or the propagation of exemplary species.
This plant is presently labelled in the garden as Ephedra americana var. andina. Most information in books (what little there is) will be under that name, though it is now treated as a synonym of Ephedra chilensis. Both names, however, hint at the current distribution of the species: the Andean (andina) mountains of Argentina and Chile (chilensis).
If you’d like to read more about Ephedra, you’ll likely find search engine results filled with commercial sites. Instead, I suggest visiting a previous entry on BPotD, Ephedra frustillata.