Kingdom:   Plantae
(Unranked):  Angiosperms
(Unranked):   Eudicots
(Unranked):   Core eudicots
Order:       Caryophyllales
Family:      Basellaceae
Genus:      Anredera
Species:     A. cordifolia
Binomial name: Anredera cordifolia
Synonyms: Boussingaultia basselloides, Boussingaultia cordifolia
Common names: Madeira vine, Mignonette vine.

Anredera cordifolia is a South American (Brazil, Bolivia, Colombia, Ecuador, Peru, Argentina, Paraguay and Uruguay) species of a succulent vine which has been introduced to New Zealand and is now considered an invasive species. It is listed on the New Zealand National Pest Plant Accord which limits its cultivation and sale.
It is a perennial evergreen climber growing up to 7 metres high. It has bright green, heart-shaped, fleshy shiny heart-shaped leaves 4–13 cm long. It produces 'warty' aerial tubers on stems and these are a key to identifying the plant.
It produces masses of small fragrant, cream flowers on dependent racemes, which may be up to 30 cm in length.

A. cordifolia can reproduce through the proliferation of tubers which detach very easily and also from rhizome fragments that may be broken off. Although this species has both male and female flowers they rarely reproduce sexually and produce seed. This species often spread through its own vegetative growth, but can easily be transported by human activities. If fragments end up in waterways, they are easily transported to new locations in this manner. It forms dense long-lived infestations that smother native plants and dominant medium to high canopy level. The combination of fleshy leaves and thick aerial tubers makes this a very heavy vine and can easily break branches and bring down entire trees on its own. It invades forest margins, rocky places and coastal areas and prefers fertile soils in warm, moist climates.

How to destroy:
Hand pull young vines or seedlings. Take care to collect all tubers, which are easily dislodged, and can regrow. Burn them or put them in a bag to rot.
Drill larger stems and squirt in 5-10mls of 5g metsulfuron-methyl (600g/kg e.g. Escort®) per 1-litre water.
Clear it off trees you want to keep and spray the vine carefully with 5g metsulfuron-methyl + 10 ml penetrant per 10 litres water.
Repeat applications are necessary until all tubers are killed.
Caution: when using any herbicide or pesticide PLEASE READ THE LABEL THOROUGHLY and follow all instructions and safety requirements.

Anredera cordifolia Madeira vine Mignonette vine. . .JPG

Anredera cordifolia Madeira vine Mignonette vine. -003.JPG

The flowers racemes.
Anredera cordifolia Madeira vine Mignonette vine. -001.JPG

Anredera cordifolia Madeira vine Mignonette vine. -002.JPG

The upper surface of a heart-shaped leaf.
Mignonette Anredera cordifolia-005.JPG

The underside of a leaf with its large raised veins.
Mignonette Anredera cordifolia-004.JPG 

The thick aerial tubers. Notice green growing tips.
Mignonette Anredera cordifolia-006.JPG

Thanks to Wikipedia for text and information: