Put ½ cup of the mint in a medium glass or stainless-steel bowl. In a small stainless-steel saucepan, bring the water to a simmer over moderate heat.
Add the sugar and stir until completely dissolved.Stir in the vinegar, garlic, and ¼ teaspoon of the salt. Continue to cook 8 or 10 more minutes, until the sauce reduces some and gets slightly syrupy.
Pour the mixture over the mint in the bowl.
Let sit 15 minutes.Light the grill or heat the broiler. Coat the tuna with the oil. Season with the remaining ½-teaspoon salt and the pepper. Cook the tuna for 4 minutes. Turn and cook until done to your taste, 3 to 4 minutes longer for medium rare.Strain the mint sauce through a sieve into a sauce boat or serving bowl. Stir in the remaining mint. Pass the sauce with the fish.
