That's right, be careful of blanket statements. You can also 'push the zone' like you do, Alex. Growing tropicals indoors in Minnesota . . . that go outdoors in summer isn't introducing invasives.
That being said, Deb's suggestions might not work too well in the Deep South but wouldn't be invasive. They'd just probably die of the heat in summer or not enough cold dormancy in winter.
Chris, we really do need to know where you are at least city, state and country, before we can recommend plants for you.
