Well, I'd still highly recommend composting. You don't have to improve the soil all at once, but most common edibles just have not evolved to grow in poor soil - they've been bred over generations and generations to be pampered a bit. I'm not sure what your motivation is for choosing fertilizer over compost, but you'll actually be kinder on your native soil by composting than by adding fertilizer.
That said, I live in a very different climate than yours (Upper Midwest) and have way more rainfall, but I do have alkaline clay. The plants that do best in areas of my soil that I've not improved yet are herbs - mints, thyme, oregano, sage, chives, winter savory and dill are all very happy without any changes to the soil. French sorrel also does great and can be used as a vegetable. Kale would be worth trying - it comes up happily in my garden paths where the soil is poor.
Beyond that, I think I'd see if there are any old-time gardeners in your area who save seed from their own gardens. You're more likely to find strains that do well in your native soils and climates if you can find seed that has been selected for it. Good luck!