Mix soy sauce, 2 tablespoons lemon juice, and half the garlic.
Cut fish into 6 or 8 equal pieces, then rinse and pat dry. Rub soy mixture all over fish. If making ahead, cover and chill up to 30 minutes.
In an 8- to 10-inch frying pan over high heat, combine oil, onion, and remaining garlic. Stir until onion begins to get limp, about 2 minutes. Add mango nectar and 3 tablespoons lemon juice; stir until boiling.
Remove onion mixture from heat. Stir in mango, cilantro, and lemon peel. Use hot or cool.
Arrange fish pieces, side by side, in a wire grilling basket. Close basket and lay it on a barbecue grill over a solid bed of hot coals or high heat on a gas grill (you can hold your hand at grill level only 2 to 3 seconds). Close lid on a gas grill.
Cook fish, turning once, until it is barely opaque and still moist-looking in thickest part (cut to test), 8 to 10 minutes total.
Serve with salsa and lemon juice.