The second-to-last post here claims that
Actually, the plant used in the movies is white thyme. It grows everywhere, can be used medicinally, is considered an invasive plant and is also given to the pigs to eat. It is also found in purple
However, the images they show are of Thymus serpyllum (Wikipedia), whereas the only references I can find to invasive thyme in NZ are about Thymus vulgarum (wild thyme) (Wikipedia), e.g. here.
The last post says it's common myrtle (Myrtus communis).
(Images to follow.)