Chop the bananas and drizzle them with lime juice to prevent browning. Set aside while you prepare the sugar mixture.
In a saucepan over medium-high heat, combine granulated sugar, brown sugar, and water. Stir until the sugars dissolve and the mixture begins to boil gently.
Stir in the bananas, vanilla extract, cinnamon, and rum. Let the mixture simmer, allowing the bananas to soften.
As the bananas cook, mash them gently with a fork or potato masher. Continue simmering for about 30 minutes, stirring occasionally, until the mixture thickens.
The jam should no longer be runny and should coat the back of a spoon. If needed, cook a little longer to reach the desired thickness.
Pour the hot jam into a clean jar and refrigerate. As it cools, it will continue to thicken into a luscious, spreadable texture.
