Cut a 4-inch square of cheesecloth. Place cloves and peppercorns in center, and form a bundle; secure with kitchen twine. Place bundle in a low-sided 6-quart saucepan. Add vinegar, red onion, red-pepper flakes, salt, and sugar. Bring to a boil, then reduce heat to medium-low and simmer 15 minutes, stirring occasionally.

Stir in ginger, lime zest and juice, and mangoes. Continue to simmer over medium until mangoes begin to soften and liquid has thickened, about 40 minutes. Remove from heat. Transfer to a large bowl set over an ice-water bath to cool; refrigerate in an airtight container up to 1 month.