Of course that “neutral third party” will act as if it were a government, but it doesn’t necessarily have to be a successor to today’s governments. It could be one successful corporation that evolves into a de facto government, and allows the other corps to exist as long as they pay tribute and don’t cause trouble. This is exactly the same role that today’s governments served.
Let’s look at how corporations came to exist in the first place. Governance is backed by force. One guy gets a bunch of his buddies together, they grab their weapons, and march over to some other guys. They kill everyone who fights, grab all their stuff, and enslave all the survivors. Or, they threaten to kill everyone, and the second group agrees to fork over regular tribute in return for not being massacred.
Then a third group comes along and wants to massacre the second group and take all their stuff and enslave the survivors. Not so fast says the first group, those are our serfs. We’ll fight you before we let you rob them, because that’s our prerogative.
And so the warriors protect the serfs from other warriors. And the head of the warriors is the king. The serfs gladly pay tribute to their masters in return for protection from even worse bastards.
And pretty soon the warriors realize that the more shit the serfs produce, the more there is for the aristocrats to steal. This is called “taxation”. The aristocrats have an interest in growing the economy, in encouraging trade and so on, rather than stealing everything all at once.
So corporations are just another way that the aristocrats encouraged the production of wealth. And people who earned wealth from this method often joined the ranks of the aristocrats, often by literally marrying into aristocratic families.
But the point of this is, corporations are not independent entities. They are money-making cooperative ventures that rely on the rule of law to exist. Without the rule of law there are no such things as property rights, only rule by force.
So with no government, why does the corporation answer to the shareholders? The wealth of the corporation doesn’t belong to the shareholders, it belongs to whoever the warriors say it belongs to.
Note that Somalia isn’t anarchic at all. You can’t just move to Somalia and do whatever you want, because it is governed by clans. Try to shoot people in Somalia and the clan leaders will send guys with rifles to kill you. Somalia is no more ungoverned than feudal Europe was ungoverned. Sure, there’s no central government, that just means that the local government–the clan–is extraordinarily powerful. If the central government was stronger the clan leaders would be afraid to straight-up murder their enemies. But with no central government the clans rule things to suit themselves.
This is the problem that people can’t seem to understand. Weak central government means strong local government of one kind or another. If the local potentates aren’t answerable to anyone, they can do as they like.