Fruiting plants to consider:
- Ring of Fire chile pepper (I grew this in 2016 in a raised bed in the shade, in crowded conditions, and it produced a nice amount of tasty, very hot peppers). I highly recommend trying this pepper. I'm guessing some other small, productive peppers might do similarly. The kinds that ripen a whole bunch of fruits pretty much all at once, such as Ring of Fire, may be best (rather than the one at a time ones, such as most, if not all, bell peppers). Ring of Fire isn't a tiny pepper, but the fruits are small (2–3 inches long).
- Grandpa's Home chile pepper (this is said to do so well with fruiting on limited light that it can make a houseplant; I also tried it in the same raised bed in the shade in crowded conditions, in 2016, and it got a decent amount of small, spicy fruits that tasted a lot like ripe Yatzy peppers). These fruits are very, very small, and they all ripen pretty much at once.
- De Barao Tsarsky tomato (See the link, which says that according to Russian websites, it's supposed to be productive, disease-resistant and shade-tolerant.)
- You might try some parthenocarpic cucumbers (e.g. Monika), but I don't know how well they'll do.
- You might try parthenocarpic tomatoes (e.g. Legend, Siletz, Santiam, Oroma, Saucy, Gold Nugget), but I don't know how well they'll do. Oroma may produce fine, depending on your growing conditions (it produced better for me in the shade at the end of the growing season; so, it may like it cooler and wetter; the rest of my season is very hot and dry; I got more fruit in the shade than in full sun, but the plant in full sun was more crowded, and in problematic soil)
- You might try parthenocarpic peppers (e.g. Planet F1), but I don't know how well they'll do in your climate. I tried Planet F1 in 2016. It got a few peppers per plant for me (nothing too special), but it faired similarly with more sun (I'm guessing it's used to a more temperate area, like the Oregon coast).
- Arugula: seems to do just fine on limited light in my experience.
- Mizuna: ditto (Arugula did better, though.)
- Shark Fin Melon: its greens grow very well and fast on limited light, although I don't know how well the plant fruits on that light. You can eat Shark Fin Melon leaves and shoots. They taste like a mix between spinach and green beans.
- Chives: They do well in partial shade. In my opinion, they're one of the best vegetables for eating (raw or cooked), and people use them way too sparingly (and not just because they're much milder than onions). These are perennials.
- Spinach (I believe I read something that said spinach can do well in low light conditions. Plus, my Grandma's perennial spinach was shaded for at least a good part of the day, and it was quite productive)
I would recommend avoiding the following (they didn't do well on limited light for me): Black Beauty zucchini, Italian Striped zucchini, Cimmaron lettuce, and Merlot lettuce (the lettuces probably would have done better in the shade with less drought, though, but I'm not sure how much better).
I would also recommend saving seeds from plants you grow in the shade. They may do better over the generations from saved seed, since they'll have a chance to acclimatize.
You might look for plant breeds that fruit well indoors (if they can do that, they can likely handle the shade).