Since my last post on this, another argument has been brought to my attention. We have, in the United Kingdom, an apolitical head of state. (Some people say that Elizabeth II is just an exception to the general tendency of monarchs to interfere in politics. But she is not so much an exception as a continuation of the development of an apolitical Crown: Most monarchs before her were similarly unprecedented in how little they interfered. It's not impossible that this will change after Prince Charles succeeds to the throne, but it's not very likely. Elizabeth II has established new standards of behaviour for monarchs. Violations of these standards are considered a betrayal of trust, and cause outcry greater than could be provoked by any politician who betrays his promises.) Moreover, the monarchy has evolved to be independent from other parts of government. A distinction is thus set up between the state and the government.
When the government has been entirely privatised, the concept of the state of the United Kingdom could still serve an important purpose. The first anarcho-capitalist society will be surrounded by nation states (we can all agree that a worldwide transition to anarchy is comparatively implausible). There will need to be a clear political boundary between it and those other states, much as we need established boundaries between states now for international law to function. The border of the ancap 'state' couldn't be completely free, because this would allow, in principle, for people to flood in from outside, be lawless, and overwhelm and destroy the ancap system. People would have to recognise the legitimacy of this border. So the monarchy would serve the same purpose in an ancap society as it does now: namely, to maintain traditions of legitimacy and to be a symbol of the sovereignty of the UK.