Put 1/3 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 1/4 teaspoon of the salt.
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 1/2 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 sauceboat or serving bowl. Stir in the remaining mint. Pass the sauce with the fish.
Fish Alternatives: Mackerel and salmon are both full-flavored enough to take on this sauce.
Wine Recommendation: Look for the lightest, leanest white you can find; anything else will have a tough time with the mint. Try a verdicchio-based wine from Italy or a vinho verde from Portugal.
