Run thumb underneath rind of ham to separate from fat. Peel and remove rind. Use a sharp knife and score fat in a diamond pattern.
Place honey, sugar, cumquats and both gingers in a saucepan over medium-high heat. Bring to the boil. Reduce heat to medium. Simmer for 5 minutes for flavours to develop. Stir in lime juice.
Place ham in a large roasting pan. Spoon over the ginger and honey mixture and secure star anise and some of the cumquats with toothpicks into the top of the ham to decorate. Bake ham for about 1 hour, basting every 15 minutes with remaining honey mixture until golden.
Meanwhile, cut potatoes into 2-3cm cubes. Cook potatoes in boiling salted water for 5 minutes to parcook. Drain well and set aside.
When ham is golden brown, remove from oven, cover and keep warm.
Increase oven temperature to 220°C. Heat duck fat in a large roasting pan in the oven for 5 minutes or until just starting to smoke. Carefully add potatoes and toss to coat. Roast, turning once or twice, until golden and crunchy. Drain off duck fat and keep potatoes hot.
To make caramel, place the caster sugar and 2 tablespoons of water in a saucepan over low heat and stir just until sugar is dissolved. Increase heat and cook without stirring until light golden caramel in colour.
Remove caramel from heat and toss immediately through potatoes until well combined. Season with salt and pepper. Transfer potatoes to a hot serving bowl.
Carefully transfer ham to a serving platter. Remove toothpicks from ham leaving star anise and cumquats in place as decoration. Decorate plate with extra fresh cumquats and serve with any remaining honey and cumquat mixture. Serve with caramelized potatoes.