Preheat the oven to 200C/400F/Gas 6.
Place the caster sugar in a 20cm/8in non-stick ovenproof frying pan. Heat gently until the sugar turns to caramel. Once the sugar has dissolved, add one and a half tablespoons of water and the rum and let it bubble.
When the caramel is golden-brown remove it from the heat and stir in the butter.
Sit the banana chunks in the caramel, packing them close together and leaving no gaps.
Roll out the puff pastry on a lightly floured work surface and sit it on top of the bananas, tucking any excess down the side of the pan. Glaze the pastry with egg wash. Transfer the pan to the oven for 20 minutes, or until the pastry is crisp and golden-brown.
Before serving, carefully pour off any excess liquid. Invert a plate over the pan and turn out the tatin.