At last I've found a name for these interesting fungi - that seem to appear in many dishes here. I've just cooked and eaten some here in my "house" along with a tomato and some bacon just moments ago. Yum!
They are called Japanese Mushrooms, Enoki or Entokitake, though probably have a few other names as well. According to Wikepedia they grow in the wild on trees, but the ones in the supermarket are cultivated and look quite different to their wild ancestors.
Purchased in clumps, they break easily and cook quickly - can be added to vegetable dishes or soups. The Internet says they are available in Australia - so I will have to look for them when I get back.
You can find more information here.