Divide the lime pieces, mint leaves, and sugar evenly among 2 (12-ounce) glasses. Using a muddler, mash the lime pieces and bruise the mint leaves while dissolving the sugar. Add enough chipped ice to each glass to fill. Add 4 ounces of rum to each glass, then use a cocktail shaker to combine and thoroughly chill each drink separately. Pour the contents of the shaker back into the glasses and top each drink with a splash of the seltzer water. Garnish the glasses with lime and mint sprigs.
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