Stir the rum and lime juice together in a large measuring cup. Poke the strawberries all over with a toothpick. Nestle them stem-side up in a 9-by-5-inch loaf pan, wedging them together as much as possible so they stand upright. Pour the rum mixture over the top and refrigerate for at least 6 hours and up to overnight.
Drain the rum mixture and save to turn into a beverage (see Cook's Note). Put the sugar on a plate and roll the strawberries in it to coat. Transfer to a serving platter and serve immediately.
To turn the leftover rum mixture into a beverage: Add the rum liquid to a pitcher with ice and mix with 1/2 cup simple syrup (or to taste), 2 cups of water, a few sprigs of mint and some lime slices.
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