Wash the fish sections and slowly bring to the boil in 1 litre of salted water with a halved, unpeeled onion, a bay leaf and peppercorns. Simmer for about 30 minutes. Wash and clean the squid.
Pour the fish stock through a fine sieve and cook the squid for 30-45 minutes. Wash the fish fillet and cut into bite-sized pieces. Thoroughly clean the mussels under cold running water (discard opened mussels).
Clean, wash and chop the greens. Dip tomatoes briefly into boiling water and rinse with cold water, peel, quarter and remove seeds. Peel and chop the remaining onion. Peel and crush 1 garlic clove.
Fry olive oil, onions and garlic until transparent. Add soup vegetables and braise briefly. Deglaze with white wine and the squid stock. Wash lemon, cut 1/2 lemon into slices, add to the soup with saffron and the remaining bay leaf, bring to the boil and cook for 5 minutes.
Add the fish pieces and tomatoes, cover and let simmer at low heat for about 10 minutes. Wash the basil, dab dry and cut into fine strips. Add basil and mussels and cook in the soup for another 5-8 minutes.
Season soup with salt and pepper. For the aioli, squeeze juice from the remaining 1/2 lemon. Peel the remaining garlic, crush it finely and add it to the egg yolk. Add 1-2 tablespoons lemon juice and salt and whip until creamy.
Add the oil in a thin stream while constantly beating and stir until thick and creamy. Season aioli with salt and pepper. Serve with fresh baguette with the fish soup.