Place the tuna steaks in a shallow dish. Drizzle the soy sauce over both sides of fish; cover and marinate in refrigerator 30 minutes.
Combine diced papaya and the next 6 ingredients (papaya through salt) in a small saucepan, and bring the mixture to a boil. Cover, reduce heat, and simmer papaya mixture 20 minutes or until the papaya is tender. Remove the chutney from heat, and keep warm.
Place tuna on a grill rack coated with cooking spray, and grill for 4 minutes on each side or until tuna is medium-rare or desired degree of doneness. Serve tuna steaks with papaya chutney.