Combine the watermelon, mango, lime juice, and honey in a food processor. Put the grated ginger in cheesecloth, a clean coffee filter, or a double thickness of paper towels and hand-squeeze the ginger juice into the processor, extracting as much liquid as possible. Add the salt and cardamom, if using. Puree until smooth.
Chill until cold, for at least 1 hour or up to 1 day. Serve cold, garnished with the mint.
Tip: For a creamier soup, add a dollop of crème fraîche to each serving. Or, for a fun appetizer, serve the soup as shooters in shot glasses. This recipe makes enough for 40 shooters.