I transplanted some lettuce into my asparagus bed this summer, around the time I stopped harvesting them in early June. It's a fertile, somewhat shaded spot that I need to keep weeded anyway, so I figure I might as well put something there that can't handle the full July sun. This was the first year I tried it but they turned out pretty well.
You can straw mulch asparagus or not, based on your priorities, just remember that the soil temperature is what triggers them to start coming up in the spring. So if you leave the soil bare in the spring, it'll warm up fast, and you'll get that delicious early asparagus when garden crops are otherwise really sparse (often around April 20 here in PA). If you mulch, you'll cut down on the weeds, but your asparagus will come up later (although it'll also stay productive later into June). Of course the slope of your land and a bunch of other factors affecting how easily you can leave soil bare for a few weeks may come into play there as well.