The solution was to buy some poppy seeds from my local grocery store and plant them. I don't know what species or cultivar the seeds are, but I can be pretty certain they will produce edible seeds. Of course, now that I'm not (overly) worried about the ethics of it, I'm a little concerned about the aesthetics. First, the germination was poor - only a few seeds germinated. On top of that, they didn't really grow where I expected them to, since we did some rearranging after I'd planted them. So after a few got weeded out, I realized that the poppies were actually coming up, mostly in the middle of my sweet woodruff (and one in the middle of the lawn).
The poppies with more competition from other plants are pretty puny, but the one pictured has a bit of space, and it is close to two feet tall. The colour is quite attractive: a pale lavender with darker spots in the middle. Unfortunately, the form of the petals is, as you can see, not very tidy. It is growing in a somewhat shady spot, and may benefit from more sun. I will try to save the seeds and plant them again next year, as the fresh seeds should germinate better.