Answer: Since you are renting and do not wish to work to improve the soil, I would suggest you use some large containers instead. You can fill them with colorful shade tolerant plants and then take the containers with you when you move. Containers are available in a wide range of prices, but larger is better when it comes to the plants' well being and ease of care.
You might enjoy impatiens for the primary flower, then add some tuberous begonias, caladiums, and coleus for impact. These three can be kept from year to year if you bring them indoors in the fall. Many houseplants also enjoy a summer outside in the shade and can be brought indoors each winter.
You could also use the perennial hostas and ornamental ivies (Hedera helix) -- both are available in an amazing range of colors, sizes and textures. Although gorwn for their foliage, they are lovely plants and easy to grow. These can be left outdoors over the winter in containers, or will actually tolerate somewhat poor soil (better soil is of course preferred) and could be transplanted when you move.
In terms of veggies for shade, there are very few that tolerate it. You could try peas, leafy greens such as spinach and chard, and herbs such as parsley and chives.
Keep in mind also that shade is a relative term. Plants receiving six hours of sun including the hour of noon are considered to be in a full sun location. Also, the sun angle changes with the seasons so an area that is quite shady in December may actually be in the sun in June. Good luck with your gardening!
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