I have fairly hot and humid growing conditions here in the American South. Broadbeans are nonexistent here even as a food item. I do grow limas and butterpeas. It's a race here on cool weather crops to get them in early enough so they have time to "make" before the heat. I start sweet peas (the flower) in the house and put them out very early and it still usually gets too hot for them and they don't do well. Still I keep growing them hoping "this will be the year the weather will be just right." Gardener's tend to be optimistic I notice. 8)
I get an idea now of the conditions--sounds challenging. I passed through Memphis on the way back to the east coast a few years ago. We stopped and walked along the promenade there next to the Mississippi. There is a sort of visitors center or something. There were people who kept asking for cigarettes there. I found that odd. I can also remember Lima beans. I developed a real taste for the so-called "Baby Limas" whilst there. I tried to grow trhem here, but that did not work at all. It is too cool and tpo moist here. As for the sweetpeas--eveb here I have to plant them in a rich compost and native ground mix and then mulch them after a while. I sometimes put newspaper and cardboard in the bottom of the hole/trench before pouring in the compost (you know, like for runner beans). It helps to hold the moisture and moisture is something that sweetpeas really like. I wonder id Broad beans would work there...I suspect that it would, for I have seen them growing is parts of Spain similiar to your weather conditions. You might want to give them a try sometime. Let me know if you do and I will tell you a trick or two. I have noticed that gardners are seemingly by natrure optimistic. I guess you'd have to be, sometimes.