Heat oil in a large saucepan over medium heat. Add onion, ginger, hot pepper and salt; cover and cook, stirring often, until soft and just starting to brown, 6 to 8 minutes. Add allspice and cook, uncovered, stirring constantly, until fragrant, about 30 seconds. Add bananas, vinegar, rum, soy sauce and turmeric; increase heat to medium-high and bring to a simmer, stirring often. Reduce heat to low and cook, stirring occasionally, until very thick, 15 to 20 minutes. Cool slightly and puree with water in a food processor or blender. (Use caution when blending hot liquids.) Serve warm or cold.