I don't know your exact growing conditions (I'm in a very different climate) but broadly speaking, leafy things will grow in shade way better than fruit-bearing things. I grow various herbs (mint, parsley) in partial-to-full shade, along with chard and warrigal greens (new zealand spinach) and sorrel and a few other bits and pieces. The chard isn't as vigorous as the stuff I grow in better light, but that might also be the soil, I'm not sure.

I've had reasonably good luck with zucchini in the shade–I once had a house that had three plants in an area with only a few hours of sunlight a day, and they were very prolific. I've also had good luck with, like Skud said, herbs–oregano, mint, and thyme have all done well for me.

Seconding the leafy-greens suggestions of chard or kale… or perhaps try some pole beans that would soon outpace the asparagus in height? Or edible flowers that would also attract helpful pollinators to your other plants… try pansies, johnny-jump-ups, borage, or oooh, nasturtiums!

Thanks for the suggestions, everyone. :)

I have a couple of shade beds, some permanent-shade and some seasonal. Still experimenting a bit with what works (we have hot dry summers and coolish mild wet winters here). The parsley is loving the permanent shade under my Tecoma tree, as is the golden oregano. My purple and variegated sage plants are also mostly coping but they don't produce the interesting colours on their leaves. I have agastache and salvias in the far extent of that bed also, placed to get sun around midsummer or midwinter depending which side of the Tecoma they're on, with mixed results. The chia plants liked the summer shade at noon (those that were placed to get it) but didn't get enough sun to make seedheads (I think sun was their main problem). Chard/silverbeet might do well - mine likes the shade it gets at midwinter and is growing beautifully for me this month, and it did benefit from light shadecloth in summer too (I recall asparagus beds not being too densely dark). If I wasn't in a hot-dry location I'd grow violets in my shady spots :-) but I'm mainly experimenting with legumes (fenugreek, chickpeas, liquorice), salvias and agastaches, and amaranth. Also with the Umbelliferae - fennel, dill, caraway and the like - which might be able to compete in height with the asparagus.

I have found ginger grows reasonably well, as does lemon balm and mint…