Preheat oven to 200°C/Gas 6.
In an ovenproof dish, mix the potatoes with the olive oil, rosemary and some seasoning. Place in the oven and roast for about 30 minutes, until the potatoes are starting to brown.
Meanwhile, prepare the artichokes (see note). Cut each one into eighths and add the segments and the wet garlic to the potatoes. Mix together and return to the oven for another 30 minutes, until the artichokes and potatoes are cooked through. Serve with the cooked broad beans.
Pull off the dark outer leaves, working around the artichoke until you are down to the pale green leaves. Then cut off the tops of the remaining leaves and trim the artichokes until no dark green bits are evident. The stem can be trimmed too – just lightly if the artichokes are young, but cut right back if old and tough. At this point, you can remove the hairy choke in the middle with a teaspoon. Once cut, the surfaces will soon brown. This can be avoided by acidulating some water with the juice of a lemon and dunking your prepared artichokes in it prior to cooking.