Heat broiler. Line a large baking sheet with aluminum foil. Toast rye bread under broiler for 3 minutes, turning once. Set aside.
In medium-size bowl, combine tuna, mayonnaise, celery, carrots, pickle relish, hot sauce, salt and pepper. Stir until blended, then taste and adjust seasonings, if needed.
Spread a scant 1/2 cup tuna salad onto each slice of rye toast. Top each with two slices of cheese. Return to broiler; heat 3 minutes, until cheese is melted and lightly browned. Serve one open-face melt with sliced tomato on the side.