I have my asparagus growing in a tangle of blackberries. In the winters, I have mowed back the blackberries and thrown down wood chips, and the following year the asparagus come up as the blackberries retake lost territory. It seems to work well.
Perhaps consider creeping vegetables like winter squash, melons, or sweet potatoes. One problem is the density and itchiness of the asparagus plants, which might hinder tending and harvest of the intercrops. Winter squash seems to rise to the top, since it offers the possibility of a one-time harvest after first frost, when the asparagus plants are dying down anyway, and harvest doesn't risk disturbing the asparagus roots, as sweet potatoes might.
If you are growing white (bleached) asparagus then there is only summer ground coverings witch you could use ( Tetragonia tetragonioides , Lotus edulis , .... ), green manure plants being the best . For green asparagus, I agree with Alder Burns. Leeks could be a late summer intercrop but don't cover the ground very effectively. Maybe lettuce could be a summer-late summer ground cover.
Let Nature work for you.
The problems of the world fade way as you eat a piece of pie. This tiny ad has never known problems: