Update: my cherry tomatoes beat the bush beans on my north facing balcony and it is not even close
I have a 4 foot by 2 foot railing box on a north facing balcony in Columbus and I ran a side by side test this summer with one Sweet 100 cherry tomato and a row of Provider bush beans. The beans got maybe 3 hours of weak sun and gave me a handful of sad pods, but the tomato still cranked out 41 fruit by mid August in the same shade. Turns out the cherry types just shrug off low light way better than beans do, at least on my railing. Has anyone found a pole or bush bean that actually produces in less than 4 hours of direct sun, or should I give up on beans and put in another tomato?
Beans are just plain sun hogs, that is the whole problem. I fought this for two seasons on my east facing porch and every bush type I tried gave me nothing but leaves. What finally worked was Rattlesnake pole beans, they still want light but they climb up and chase it, so the top of the vine gets sun even when the base is shaded. The trade off is they take forever, like 80 days, so you are picking in September while your tomatoes are already done. If your balcony really only gets 3 hours, I would stop fighting it and run two cherry tomatoes in that box instead, maybe a Sungold next to the Sweet 100 for some variety.