To make the raspberry syrup, combine the the sugar, 1/2 cup of water (use 1 cup if using 1 cup sugar), and 1 cup of raspberries in a sauce pan. Bring to a simmer and heat for 5 minutes, crushing the raspberries with a spoon to release the juices. Let cool, then strain out the solids through a fine-mesh sieve.
In a large pitcher, pour in the wine, rum, lime juice, raspberry syrup, and raspberry jam. Add in the remaining 1 cup of raspberries and the sliced lime. Stir to combine.
Cover and chill the sangria in the fridge at least two hours. You may need to stir the sangria to redistribute the jam. Then, pour into glasses and top with sparkling water and fresh mint.
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