Wash and dry the limes, finely grate the peel. Squeeze the limes. Cream fat, 250 g sugar, lime zest of 1 lime and salt with the whisk of the hand mixer. Stir in eggs one after the other. Mix flour and baking powder. Alternately stir the milk, juice of 1 lime and 4 tbsp. gin into the fat cream.
Place in a well-greased, flour-scattered box mould (approx. 2 litres capacity, 30 x 11.5 cm) and smooth down. Bake in the preheated oven (electric: 175 °C/circulating air: 150 °C/gas: see manufacturer) on the middle shelf for 50-55 minutes. Cover if necessary approx. 10 minutes before the end of the baking time.
10 minutes before the end of the baking time, place 75 g sugar, 75 ml gin, tonic water and the remaining lime juice in a saucepan. Heat, bring to the boil and simmer for about 2-3 minutes. Remove the cake from the oven, prick several times with a wooden skewer. Pour syrup evenly onto the cake. Let it cool down a bit, carefully turn over on a grid, let it cool down.
For the icing, mix icing sugar, sour cream and 1 tablespoon of gin to a thick, smooth icing. Spread on the cake and sprinkle with the remaining lime zest.