Although you can't see evidence of slugs, my guess is that slugs are nibbling at your asparagus. Tiny slugs often don't leave much of a trail. The bent stems are a good clue - the slug has a nibble at one side, and the stem then distorts as it grows. Sometimes rots can do this, but if the stem is good and firm, it's probably slugs. A dry mulch will help deter them, with a few organic slug pellets if you want. I haven't noticed woodlice attacking asparagus here.
Problem I've had this year is ground frost knocking out quite a few spears, this weather is hard to deal with.
