Peel the fruit. Cut away the membranes of the individual segments with a sharp knife. Collect the peeled segments in a bowl.
Drain off any excess juice from the fruit into a small saucepan. Add the lime juice, honey, and cardamom to the saucepan. Bring to a boil and reduce heat to low. Simmer for 10 minutes and then remove from heat and let cool to room temperature Pour over fruit mixture and gently fold in so that all the fruit is coated. Let stand for 15 minutes or chill until ready to serve.