Whisk the marinade ingredients together in a bowl. Set aside 3 tablespoons for the relish and brushing.
Arrange the tuna in a baking dish. Pour the remaining marinade over the fish and turn to coat. Refrigerate for 30 minutes to 1 hour, turning once or twice.
Combine the relish ingredients in a bowl. Add 1 tablespoon of the reserved marinade and stir to blend. Refrigerate until use.
Prepare the grill for direct cooking over high heat (450° to 500°F). Preheat the sear grate over direct high heat for at least 10 minutes.
Prepare the grill for high heat (450° to 500°F). Preheat the sear grate over direct high heat for at least 10 minutes.
Remove the tuna from the marinade, letting any excess drip back into the dish. (Discard the marinade.) Place the tuna on the sear grate and cook, with the lid open, until just turning opaque throughout (or your desired doneness), about 4 minutes for medium-rare, turning once. Remove from the grill and brush with the remaining reserved marinade. Serve immediately with the relish spooned on top.