In a large bowl, combine garlic, dried oregano, kosher salt, black pepper, orange juice, lime juice, and white distilled vinegar; add chicken thighs and drumsticks and toss to coat. Cover and refrigerate 30 minutes to 2 hours.
Lift the pieces from the bowl, letting excess liquid drip back; reserve the liquid.
Heat neutral oil in a heavy pot over medium-high heat until shimmering, about 2 minutes. Sprinkle in granulated sugar and cook, stirring, until it melts and turns deep amber, 1–2 minutes.
Carefully add the pieces to the pot and brown, turning occasionally, until lightly colored on all sides, 6–8 minutes.
Add onion and cubanelle pepper; cook, stirring, until softened, 3–4 minutes.
Stir in tomato paste and cook until slightly darkened, about 1 minute.
Pour in the reserved liquid, add pitted green olives, capers, and cilantro, and stir to combine.
Add water, bring to a boil while scraping up browned bits, then reduce heat to medium-low, cover, and simmer until the pieces are tender and cooked through (juices run clear and reach 165°F at the bone), 25–35 minutes.
Uncover and simmer over medium heat, stirring occasionally, until the sauce reduces to a glossy consistency that coats the back of a spoon, 5–10 minutes. Taste and adjust seasoning if needed, then serve hot with white rice and beans.