Preheat oven to 170°C/150°C fan-forced. Grease and line a 22cm (base) round spring-form cake pan with baking paper.
Using electric mixer, cream butter, sugar and vanilla on high until light and fluffy. Gradually beat in eggs until combined. Using a large metal spoon, gently fold in coconut, flour, milk, yoghurt and lime zest until well combined. Pour mixture into pan and smooth surface. Bake for 45-50 minutes until golden and cooked through when tested with a skewer. Stand in pan for 10 minutes then transfer to a wire rack to slightly cool.
Meanwhile, to make the lime syrup, using a vegetable peeler, remove rind from 2 limes then trim off white pith. Slice rind into thin strips. Juice all 4 limes. Combine ⅔ cup lime juice, lime rind, sugar and water in a small heavy-based saucepan. Stir over medium heat until sugar dissolves. Bring to the boil then reduce heat to medium and simmer for 4-5 minutes until syrupy.
Place cake onto a large serving plate. Gently prick cake with a skewer then slowly drizzle over the hot lime syrup. Set aside to cool completely. Slice and serve with thick natural yoghurt.
