For the herb and garlic sauce, whizz all the ingredients in a mini food processor or in a jug with a stick blender. Taste and season with salt and a little pepper, then spoon into one large or 4 small bowls and chill until needed.
For the tempura, heat the oil for deep-frying to 190°C. Make the batter by sifting the flour, cornflour and a small pinch of salt into a mixing bowl. Scoop 3 tbsp into a small bowl. Quickly stir the mineral water into the remainder, until only just mixed in but still a bit lumpy, then sprinkle the reserved flour on top which helps give a light, crunchy batter.
Cut or snap off the woody ends (and discard) from each asparagus spear, then cut the spears in half lengthways. Drop the spears into the batter, about 5 at a time, so as not to overcrowd the bowl and pan. Lift them out using tongs, then quickly drop them into the hot oil. Deep-fry for about 90 seconds until crisp and very lightly golden, then remove the spears with a slotted spoon and drain briefly on plenty of kitchen paper.
Pile the tempura asparagus onto warm plates, sprinkle with the chopped herbs and serve straightaway, with the sauce for dipping.