анонимный вопрос
Answering this question is not just simple, but very simple. If people only apply to a court voluntarily, and the court has no means of coercing the execution of its decisions—which is precisely the kind of pathetic existence courts might lead under ancap—then courts have no options left but to follow libertarian principles of law.
Alice, based on the principle of self-ownership, voluntarily turned to a court to resolve a conflict with Bob. The court denies Alice self-ownership and issues a verdict stating that she has no right to own any property, so that Bob has the right of exclusive disposal over all disputed property, starting with Alice herself. Bob, in accordance with the court’s decision, appropriates all the property, including Alice, whom he keeps on a leash and beats for any disobedience.

Under ancap, there cannot be a sad girl on a leash!
What will stop Alice from sabotaging the court’s decision and actively resisting Bob, seeking any outside help? What will stop any outsider who, seeing a violation of the NAP, asks what is going on, is told it is the execution of a court decision, and responds that he doesn’t give a fuck?
Since the question itself postulates that it is ancap all around, it means the vast majority of relationships in this society are voluntary and market-based. This is precisely what everyone perceives as the norm, and any deviation from the norm must be hidden or very carefully justified.
One can ask “what would be the typical reaction under ancap to such-and-such a deviation from libertarian principles,” and in response, those typical reactions will be described, demonstrating the feedback mechanisms in an anarcho-capitalist society that make deviating from basic principles unprofitable. But it is pointless to frame the question as “how to fight mass deviations of courts from libertarian principles under ancap?” If they are mass deviations, it is no longer ancap, and one must look into why they are mass and which feedback mechanisms were lacking.
The question of whether a pure anarcho-capitalist society is stable has no theoretical solution today. Societies considered precursors to ancap in legal terms (Ireland, Iceland) were historically defeated by external forces, but this is not an argument for internal instability.