I'm in zone 8a, Birmingham, Alabama.
Invasive and difficult to get rid of:
Arum italicum 'Marmoratum'
Japanese climbing fern (Lygodium japonicum)
Alstroemeria psittacina
Spreading but controllable:
Nandina domestica
Callicarpa americana
Miscanthus (many)
Physostegia virginiana
Pennisetums
Some alliums
Creeping Jenny
Liriope
Ruellia and Turks Cap (Malvaviscus arboreus) are not a problem as I have them in full sun and never water them.
Hope this helps,
Susan