Agua de Alfalfa is very common in central Mexico, just the leaves and water (in your case ice) and a bit of sugar if desired. And it looks just like your picture, I think it would be a bit easier for you to get than chaya.
You can add a bit of limon (don't need to peel it, just cut it in small chunks before blending), and would be very refreshing.
-----Also called Lucerne----
Perennial tall herbaceous plant belonging to the pea family. It is native to Europe and Asia and has spikes of small purple flowers in late summer. It is now a major fodder crop, commonly processed into hay, meal, or silage. Alfalfa sprouts, the sprouted seeds, have become a popular salad ingredient. (Medicago sativa, family Leguminosae.)