Those are spider plants, Chlorophytum. The stalks are stolons which carry both flowers and baby plants. They are meant to trail so they touch the ground and thus allow the babies to root.

Really, with thick, stiff leaves similar to a bromeliad?

Yes.

That's definitely a spider plant. I have that same variety in my yard. Callisia fragrans runners normally emerge from along the plants main stem which can grow two feet or more in length . Callisia fragrans also doesn't flower along its runner the way spider plants do. It produces instead a separate upright inflorescence with tightly held and very fragrant flowers. I've just recently learned that callisia fragrans is edible and that it's used homeopathically to support the immune system.

I agree with spider plant, and with Florida Joes on the Callisia fragrans. Spider plant leaves get thicker and tougher when exposed to more sun. I have the variegated Callisia fragrans, and I also have spider plant. This is definitely spider plant.