In a large bowl, combine rice, cardamom, and 3 cups water. Cover, refrigerate, and let rice soak until starch is released and grains have softened slightly, at least 8 hours or up to 12 hours.
Set a fine-mesh strainer over a large bowl. Transfer soaked rice and soaking liquid to blender and blend on high until completely smooth, about 2 minutes. Pour mixture through prepared strainer, scraping bottom of strainer with spoon or rubber spatula to help liquid pass through. Discard solids.
Rinse strainer, then line it with cheesecloth and set it over a clean large bowl. Pour strained rice liquid through prepared strainer, scraping bottom of strainer to help liquid pass through. Discard any remaining solids. You should have about 2 ½ cups rice liquid.
Rinse blender. Add mango, condensed milk, coconut milk, lime juice, vanilla, salt, remaining 1 cup water, and strained rice liquid. Process until completely smooth, about 2 minutes. Transfer to a pitcher and refrigerate until very cold, about 1 hour.
Stir before serving. Pour over ice. Top with fresh mango and toasted coconut, if using. Serve.