Break up the frozen cabbage with your hands and drop in a blender.
Add 1½ cups water to start and puree on high power until it looks like very coarse shaved ice. Add up to ½ cup more water to get a thinner consistency, if needed.
Next, add strawberries, banana, and Greek yogurt and blend again until smooth. Leftovers may be stored in the refrigerator for up to 24 hours.