No personal experience with this one, but now I'm wondering if it's the 'stickers' the children were complaining about at my SILs beach house yesterday....

Oregon State U Extension "If green burs carrying seed are present, dispose of plants, preferably by burning. Seed matures continually from May-Oct and remains viable in the soil for years.

Spraying mature plant with herbicide may increase infestation when seed heads drop and are spread"

Your impression of how Roundup works is correct. I have no personal experience with this plant, but I googled it and found this link. They say that Roundup doesn't kill them; in fact, it creates the conditions that they like by killing competing plants.

There are very few plants that Roundup doesn't kill. I hope you will let us know what you find out.

Here is a link that might be useful: puncture vine

Roundup won't kill ripe seed, as this is not an actively growing part of the plant, and therefore the seed will not receive the translocated herbicide to kill it. If all the seed were still ripening and immature, it would get the Roundup into the system. For ripe weed seeds of any variety, Roundup will only work once it has germinated and is actively in growth mode. You may need to use a pre-emergent weed killer as well, which will limit what else you can plant from seed.

Often when I am dealing with pernicious weeds that will reseed, I will water to germinate new weeds between treatments of Roundup, and repeat this process several times. I prefer not to use pre-emergents, as many seem to also negatively affect growth of ornamentals.

I remember goatheads from my childhead in Phoenix, didn't know they grew here. Nasty on your feet.....

They also post here.