Heat oil in a large saucepan over medium heat. Add red pepper flakes; cook until they begin to sizzle. Stir in onion. Reduce heat to low, cover, and cook, stirring occasionally, until onion has softened, about 20 minutes.

Increase heat to medium. Add bell pepper and ginger; cook and stir, uncovered, until ginger is fragrant, 2 to 3 minutes. Stir in mangoes, pineapple, brown sugar, vinegar, and curry powder; bring to a simmer and cook for 30 minutes, stirring occasionally.

Cool chutney completely. Store in airtight containers in the refrigerator.