"From: "simy1"

Oxalis does not spread by rhizome, only by seed. If the ground covers
were thicker, oxalis would be reduced. You could also consider adding
wood chips by hand in the ground cover. That will be a total pain but
it will give you a few years of respite."

Thanks for your advice -

I just got an ID from my local farm bureau, and my oxalis is "oxalis
pes-caprae". You can see it by doing a Google Image search for Bermuda
Buttercup. According to online sources, it reproduces primarily by bulbs
and less so by seed. Nevertheless, they give me little hope short of
using pre-emergent twice in the fall. I don't wish to follow the chemical
route, so I plan to replace my aptenia (red apple) , myoporum, and miniature
ice plant, with a more robust and competitive plant. So far, my prostrata
rosemary seems to to the trick and overcome the oxalis, especially when I
use landscape fabric for the small seedlings.