Here is my chasmanthe flower. I don't have a picture of a seedling but they look a good bit like freesias or Sparaxis.
We get nut sedge in the garden, too. I pull it out when I see it but haven't gotten rid of it all. Another thing that is really invasive in the yard is vinca. It came with the house and keeps on spreading. It is now the 'ground cover' in the rose garden, where it isn't too much of a pain, but it has gotten into the vegetable garden. Maybe the vinca and nut sedge can duke it out in the garden. ;-) The vegetable garden is also full of Bermuda grass. The garden was once a Bermuda grass lawn. We've tried to get rid of it but it keeps coming back. I just live it now and pull it out in the planting holes and figure that it keeps the paths from being muddy. It's kine of weird to have to mow the vegetable garden, but that's the way it is. (It's just a small plot. We just have a '60s tract home lot.) Most everything that is invasive in our yard I've seen naturalized in the wild, though not necessarily really invasive. I'm sure the dry season keeps it in check. My favorite naturalized things to come across are English walnut and Iris pallida. Neither is invasive here. I enjoy seeing these remnants of the past.
Out in the wild, mustard and wild oats are probably the most wide spread invasives and they seem to be here to stay. Grazing or lack of grazing and fire or lack of fire doesn't seem to faze them as far as I can see. We have yellow star thistle, too, but it isn't everywhere. I'd say Italian thistle is more widespread along trails and fire roads. Milk thistle is bad in places, too.