From our fungi guru Heino Lepp:
It's Ileodictyon gracile. Clathrus ruber has red arms. If you see a species name with 'rub' in it and assume there's some connection with redness, you'll be right most of the time. Undoubtedly, there will be exceptions, but I can't think of any at the moment. Similar to Ileodictyon gracile is Ileodictyon cibarius. Both have that network of white arms and the naked-eye differences are:
Ileodictyon gracile - the arms are smooth and broaden at the junctions.
Ileodictyon cibarius - the arms are a little wrinkled or corrugated and do not broaden at the junctions.
Stinkhorns vary greatly in shape and colour but all have two things in common. First, they start out somewhat egg-like, enveloped in a gelatinous layer within a white membrane. At maturity the fungus breaks through the surrounding membrane and expands to its full size and shape.
The second thing they all have in common is that their spores are in a typically khaki to brownish slime somewhere on their surfaces. You'll have noticed the brownish muck over the white arms of your fungus. Stinkhorn spore slime smells, typically with an odour of dung or rotting meat and so attracts flies and other carrion- or dung-loving insects to disperse the spores. In my experience Ileodictyon spore slime has the weakest smell amongst the stinkhorns. Once the spores have gone so has the smell. Many, but not all, stinkhorn species have red colours - which gives an additional visual cue since it can give an impression of freshly exposed meat to a carrion-loving insect.
The first description of Ileodictyon gracile was published in 1845, based on a specimen collected in what is now the Perth area. I attach a copy of the accompanying illustration in that paper of 1845. Also, I've attached a map showing from where in Australia there are databased herbarium specimens of this species.