Juice the pomegranates by pressing them in a citrus press, using a citrus juicer or by releasing all the flesh from the membrane and pulsing it in a blender before sieving out the pips.
Dissolve the caster sugar in 2 tbsp water and then simmer it until you have a very thick, sticky syrup, then stir in the pomegranate juice to make fresh grenadine. Cool, then cover with clingfilm. This will keep in the fridge for up to 5 days.
Crack the sugar lumps in half (or into large pieces). Drop a piece each into 6 champagne glasses, then add a measure of grenadine and a slice of lime to each one. Top up with champagne (it will fizz up quite a lot because of the sugar) and serve.