Stir together rum, lime juice, soy sauce, and sugar in a shallow bowl until sugar has dissolved.
Add chicken and stir to coat. Marinate at room temperature 25 minutes.
While chicken finishes marinating, heat 1 inch oil in a deep 12-inch heavy skillet over medium-high heat until it shimmers.
Meanwhile, whisk together flour, paprika, and 1/2 teaspoon salt in a shallow bowl.
Drain chicken and pat dry. Dredge in flour, shaking off excess, then transfer to a plate.
Fry chicken in 3 batches, turning occasionally, until deep golden brown and cooked through, 6 to 7 minutes per batch. (If chicken darkens too quickly, reduce heat.)
