Put ginger and mint leaves in a clean widemouthed 1-qt. canning jar. Bruise them with a wooden potato masher or pestle (if you have one; or use a wooden spoon) until very fragrant. Add pineapple and enough vinegar to cover. Close jar tightly and shake vigorously for 10 seconds. "This gently macerates the fruit, releasing the flavors into the vinegar," says Karlin. Push pineapple, mint, and ginger down so they're completely covered by vinegar and it can draw out their flavor. (It's also important to keep the fruit completely submerged to prevent undesirable bacteria or mold from growing on it.)
Replace lid with a piece of cheesecloth (so air can get in and help start fermentation) and securely attach with a rubber band or the jar ring. Leave at room temperature about 12 hours.
Discard cheesecloth. Replace lid and ring, secure tightly, and repeat the shaking daily for 3 days, keeping at room temperature. "I usually stick a Post-it on the jar that allows me to keep track of where I am in the process," says Karlin.
Pour mixture into a wide strainer set over a bowl. Discard mint, then return solids and liquid to jar. Shake well each day for 4 more days at room temperature. "There's not much of a visual indicator, but it will be done after 7 days total," Karlin assures.
Strain out solids over a bowl and set aside for Pineapple Salsa, or another use. Return liquid to jar, using a funnel. Add lime juice and sugar, cover, and shake until dissolved, about 2 minutes. Add the mint sprig.
Chill 7 days for flavors to meld, giving the bottle a good shake every day to be sure sugar is dissolved. Discard mint. Use, or chill up to 4 months (shake before using). The shrub will get more syrupy with time.
*Find raw, unfiltered cider vinegar at many grocery stores and natural-foods stores, and raw coconut vinegar at natural-foods stores (see a store locator at coconutsecret.com). Karlin likes Trader Joe's raw organic evaporated cane juice sugar.
Make ahead: Up to 4 months, chilled.
*To make supremes, cut off outer peel and membrane of fruit, then cut between inner membranes to free segments.