Perez's Sea Lavender (Limonium perezii)
is such a beautiful plant. It's not a North American native and it is listed as vulnerable in it's native habitat of the Canary Islands. I couldn't find any reference of it being threatened anywhere else and it has apparently become naturalized and is considered a potentially invasive weed in Australia. This Australian site says that it has become invasive in California: https://invasives.org.au/blog/...
It sounds like it's become a pest plant in your garden; you could try digging them out but it might be difficult getting all of the underground rhizomes and you'd eventually have new plants sprouting.
Hopefully members from your area will be along soon with suggestions.