According to the Bushman's Friend, a great resource, Coprosma fruits are a 'succulent globose drupe'. How could your mouth not water? (Botanist-speak can be so evocative!)
Coprosma is a genus containing over 100 species of tree and shrub, most with edible berries. New Zealand, Australia, and other places around the Pacific all have their own native Coprosmas, and the berries have generally been well known to the indigenous people in these areas.
Around Wellington, two of the most common Coprosma are Karamu and Taupata.
Taupata is perhaps most instantly recognisable by its strong, super-shiny leaves. Clearly it's well adapted to withstand salty coastal wind, but it grows quite far inland as well.
Coprosma berries often have a high ratio of seed to flesh (certainly Karamu and Taupata do), so while it's fun to pick them off the tree and nibble on them, the best use of them is in cooking - where you can boil them up with a little water and some sugar to taste, and then push the result through a sieve to make a sauce for either sweet or savoury purposes!
You'll need to get quite a few handfuls to make it worthwhile.
Of course, make sure you know for sure that it's Coprosma. For help with ID'ing, google images for Taupata and Karamu is useful, but a book packed with photos like JT Salmon's Native Trees of New Zealand is invaluable.
A cup of coprosma, anyone?
In 1886, settler JC Crawford was hopeful about the possibilities for a coffee substitute industry in New Zealand, based on Coprosma seeds. The Coprosma genus is in the same family as coffee plants.
It never quite took off, but over the subsequent decades various NZ foragers like Andrew Crowe and Sheila Natusch have tried making coffee from Coprosma seeds, and been underwhelmed by the results.