I have a small bed of aspargus (6-8 plants planted last year), unfortunately it got infested with the worst weed I know; quackgrass.
I'm at a point where I wonder if I should just kill the whole bed with a tarp, sacrificing the asparagus. Quackgrass can hardly be pulled in a way that removes all its rhizomes. I probably can't move the asparagus elsewhere, because I would likely bring the grass with it. Using plastic mulch seems impractical given that the asparagus pops up randomly (both in timing and location). I probably can't stack enough mulch that would defeat the quackgrass and yet let the asparagus live. I don't live sufficiently close to that bed that I could just pluck away the grass every day I see it.
Quackgrass rhizomes are much easier to dig/pull out first thing in spring, before they start growing and send out all their fine root hairs. I'm also finding the same to be true during drought conditions (sigh).
I had a similar problem with an asparagus bed that I inherited at the community garden and found the easiest solution was to wait until all the fronds had totally died back and mulched with a really deep layer of woodchips, coffee grounds, horse manure, chicken manure, leaves - everything and anything I could find.
The couch grass is still there but as Douglas mentioned the rhizomes are much easier to pull up when they grow through the mulch.
The asparagus has no problem growing through the mulch and as an added bonus, I also had an excellent harvest of stropharia rugosoannulata that came in with the woodchips.