That is Asparagus asparagoides, a vigorous rambler from South Africa. It has naturalized and become an invasive weed in other places in the world (such as California, Australia, New Zealand), after being introduced as an ornamental/landscape plant and florists plant.
Identifying features include:
- The 'leaves' (technically called phylloclades), they are actually flattened stem segments, and show the parallel venation typical of monocots.
- Rambling/climbing growth habit
- Underground tubers (they will be much better developed under bright lighting conditions on mature plants)
- Small white flowers, followed by berries (again, more commonly on mature plants in plenty of sun).
Depending were you live, this may be considered an invasive weed in your area. If it begins to flower and fruit, I'd recommend removing it, so that it doesn't begin moving into the surrounding neighborhood.