Hello Jerome,
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.
I posted photos in this
thread
here
I will have to repeat the process annually but mulching is easier than digging up the asparagus.