In a medium stockpot, toss the apples, pears, kumquats together with the lemon juice. To the pot add everything up to and including the turbinado sugar. Bring to boil over medium-high heat. Reduce heat to low and simmer for 10 minutes, or until the fruit is tender but not mushy.
With a slotted spoon or spider strainer, scoop out the fruit and place on a cookie sheet to cool.
Add prunes and apricots to the liquid and let steep over low heat for a few minutes until plump but not mushy. Sometimes dried prunes and apricots are super dry or super wet, so adjust your steeping time accordingly. Remove the prunes and apricots and add to the rest of the cooling fruit.
Add the star anise to the liquid and bring to a boil. Reduce by about half or until slightly syrupy, about 10 minutes. Turn off heat. Remove and discard the star anise. Spoon in the grapefruit and star fruit pieces just to coat in the syrup and remove these to the rest of the fruit.
If you have one, get a beautiful clear glass bowl ready and place the fruit inside. Pour some of the syrup over top, reserving some for a small pitcher for guests to pour themselves, if there is a lot of syrup. Garnish the fruit with the ginger and extra star anise.
Serve alongside bowls of Greek yogurt, crème fraiche, some lightly spiked whipped cream, or ice cream. Also nice with baked ham and pork roasts.