Root Vegetables that are shade tolerant include ...
Beets which need about 5 hours of sunlight daily.
They do Okay in partial shade, but the harvest is delayed somewhat. They mature faster with a tad more sun.
Many Beans, legumes and peas will thrive in a shaded area requiring roughly only 4 to 5 hours of light only.
Herbs, particularly leafy herbs can be grown in sun starved corners
Ginger which is grown primarily for its roots is a perennial that will do well in shaded areas.
Also consider Ginseng, a crop that requires several seasons to reach maturity and thrives in very low light.
Fruits that are shade tolerant include several types of berries.
Gooseberries and Currants
Some varieties of strawberries and raspberries.
will not do exceptionally well in full shade,they need about 6 hours of sunlight daily which makes them tolerant of partial shade. Strawberries grown in partial shade should be planted earlier than normal to allow more time to concentrate their sugars. Alpine Strawberries are the best so far as shade tolerance is concerned, the berries produced in plants grown are shade are inferior in quality to those receiving full sun but they are acceptable. Musk strawberries are also shade tolerant. They are a tad different from most garden varieties of strawberry, they are also not self pollinating and you need a male and a female plant to achieve fertilization.
Raspberries, like strawberries will not produce as well in a shaded area - but they are tolerant of partial shade. They are actually more tolerant of partial shade than they are of full intense sunlight for extended periods.
Huckleberries are similar to blueberries and many will grow in the shade just fine. Black huckleberry and evergreen huckleberry shrubs are the best for shade tolerance.
Gooseberries and currants will grow in partial shade but full sun will yield more fruit.
Mushrooms do better in shade and they are not all that difficult to grow, but do require a little patience.
Tips for Gardening in the Shade
Sunlight and warmth facilitate evaporation, so naturally a shaded area will tend to be moister for longer periods than a sun drenched spot. Plants growing in the shade due to their longer exposure to moist conditions tend to be more prone to fungus, mold, mildew and related diseases. Take appropriate measures to guard against these problems.
Soil should be well drained, heavy clay soils and soils that retain excessive moisture are not a good thing. They are a harbinger of botanical failure. While it is a fact that soil conditions will vary widely in any locations, shaded location soil may need to be conditioned to ensure it is well draining and has organically rich amendments. Leaf debris, composted leaves, leaf mold should be present in abundance, if it is not present it is easily acquired and can be used as a mulch as well as being worked into he soil.
Pruning. If your shade is overly dense, it is advisable that you trim the trees that are causing the excessive shade. Remove low hanging branches and, thin out sections with thick canopies.
Reflective Mulch is another good idea for shade gardens. It makes optimal use of available sunlight by reflecting it back to the plants. Metallic mulches, red plastic mulch for certain plants. Metallic mulches such as Mylar which is used in extensively in hydroponic gardening can be useful outdoors as well, they look like aluminum foil, but are tougher. Other reflective surfaces in the garden vicinity are also beneficial to plants. A brightly painted fence facing the sun and reflecting light onto the crops is ideal.