Heat the oil in a large saucepan, add the onion, potatoes and chorizo and fry for 5 minutes, stirring regularly until just beginning to soften.
Stir in the tomatoes, broad beans and stock, season and bring to the boil. Cover and simmer for 15 minutes until the vegetables are tender.
Mash some of the potatoes roughly with a fork to thicken the soup slightly. Taste and adjust the seasoning if needed.
Toast the bread on both sides then rub one side with the halves of garlic and spread with the tapenade. Ladle the soup into bowls and top with the tapenade toasts and sprinkle with basil leaves to garnish.