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Put the blackberries in pan with 4 tbsp of the sugar, add a splash of water then cook until the berries have softened. Rub through a sieve. Cool then freeze in a plastic container until slushy, stirring every now and then.
Whizz the mango with 75ml of its syrup and 2 tbsp icing sugar until smooth, then freeze in a container until slushy as above.
Stir the rest of the sugar with the lime zest and juice. Beat the cream with 3 tbsp of the remaining mango syrup until it forms soft peaks, then beat in the lime mixture. Freeze in a container until semi-frozen.
Line a medium Pyrex basin with cling film then beat all three icy mixtures and spoon alternately into the bowl to make rippled layers. Return to the freezer until solid.
To serve, pull out of the bowl and strip off the cling film. Put on a plate, scatter with the crushed biscuits and leave for 30 mins to soften in the fridge before eating.