Can you please post an image so that I can see if you have Myrsine aquilonia? This species is common in cultivation around Auckland now, and it does occasionally set viable seed and self establish - but I have not seen that very often. However, at Auckland University M. divaricata (in the strict sense) does naturalise freely, and that species (in the strict sense) has a superficial similarity to M. aquilonia (its quite different from the usual form of M. divaricata people know and refer that name to).