Cut red snapper into 1/2-inch cubes and place in glass or ceramic dish. Toss with 1/2 cup of lime juice. Cover and marinate in refrigerator 15 minutes. Drain and discard lime juice.
Transfer fish to medium bowl. Stir in remaining 1/4 cup lime juice, tomato, onion, cilantro, clam juice, chiles, and salt. Chill at least an hour, or up to 1 day, for flavors to blend.
Serve in tall chilled goblets with spears of romaine lettuce and/or tortilla chips, or in soup bowls lined with the lettuce leaves.
Other Notes
Romaine lettuce leaves for garnish
Tortilla chips for garnish