Place beet and garlic in the blender and process to a puree.
Add ice cubes and process until the ice is completely crushed.
Add water, coconut milk, salt, lemon zest, spices, and rum and process for a few more seconds; check the thickness and add water if necessary. Taste and adjust seasoning if needed.
Pour into wine glasses or soup bowls, gently drizzle with vegan sour cream or yogurt, and sprinkle with sumac and grated lemon peel. Serve with beet chips or beet crackers.