Heat oil in medium saucepan on medium heat. Add onion; cook and stir 5 minutes or until softened. Add garlic, jalapeño, turmeric and allspice; cook and stir 30 seconds or until fragrant.
Stir in bananas, vinegar, sugar and sea salt. Bring to simmer on medium heat. Reduce heat to low; cover and cook 15 minutes, stirring occasionally. Remove from heat. Let cool 10 minutes.
Spoon mixture into blender or food processor; cover. Blend on high 1 minute or until smooth. With center part of cover removed and blender on, slowly pour in water and blend until the consistency is similar to traditional ketchup.