In a medium saucepan over medium-high heat, cook the sugar to a light amber color. (Check the color by drizzling some onto a white plate.) Don’t worry if the sugar hardens while cooking; simply stir it and continue cooking until it smooths out.

Add the blueberries and ½ cup water to the caramel; be careful, as the caramel will bubble and steam a great deal. Stir the blueberries into the caramel as well as possible (the sauce will harden when the cold blueberries are added, but will loosen up again as the blueberries heat up). Cover the pan and reduce the heat to low. Allow the sauce to simmer with the lid on until most of the blueberries have popped and are exuding their juices, about 5 minutes. Remove the lid and stir the sauce to further dissolve any hardened bits of caramel. (Don’t worry if there are some lumps; they will be strained out.)

Strain the sauce to remove any blueberry skins, seeds, and hard pieces of caramel. Add the lime juice and return to a saucepan. Cook further over medium heat until the sauce can be spooned onto a room-temperature plate and keep its shape without being too runny. Serve at room temperature.