In a large, heavy-bottomed pot, cover fruit and star anise with 4 cups water. Bring to a strong simmer over medium–high heat. Cook until water is reduced by about 3/4, and fruit is soft and tender, 20 to 30 minutes.

Add sugar and continue to cook, stirring occasionally at first and more frequently as the jam cooks and juices thicken until most of the liquid has evaporated and fruit has started to break down, about 30 minutes.

As jam cooks, you’ll notice the liquid go from a rapid, rolling boil with smaller bubbles to a slow, thick, tarlike boil with larger bubbles: Stir constantly along the bottom of the pot to prevent scorching and sticking. (Sugar is heavier than water and will concentrate there, increasing any chance of the fruit burning.) It’s also the stage at which splattering may occur, so take care in stirring.

When jam reaches a slow, thick boil, add lemon juice plus any add-ins and continue to cook, stirring constantly until jam has returned to its previously thickened state, about another 5 minutes. Using a spoon or other utensil, pick out the star anise. Divide between jars, leaving 1/4 inch of headspace, and seal immediately.

Store in the refrigerator and use them up within a couple of weeks.