In an anarcho capitalist society, all law will be what was known in medieval times as private law.
This unavoidably has the effect that in general people will not be equal before the law.
People who can pay well, people who can defend themselves well, and people who defense agencies believe are unlikely to create trouble will obtain rules that favor them
People who are poor, people who are incapable of defending themselves, and people who defense agencies believe are likely to create trouble will face rules that disfavor them.
Respectable middle class people will be treated before the law about as unequally as credit card companies treat them today, which is to say that one respectable middle class person who pays his bills is treated by the credit card company in a way that is is similar but not identical the way it treats another respectable middle class person who pays his bills, and very different from the way it treats the broke person who frequently fails to pay his bills.
In medieval times, access to the market in defense and arbitration services was severely unequal, because of the severely oligopolistic nature of that market, and as a result private law treated people grossly unequally. Thus the phrase “private law” became “privilege” (“privy lege”)
I would expect that in a genuinely free market for defence and justice, where access to defense and arbitration was roughly equal, law would similarly be roughly equal.
This reflects the fact that a rich man's estate is subject to greater threats, in proportion to his ability to pay for defense against those threats. Indeed since a rich man is less capable of himself defending his property than a poor man, it is likely that rights to property directly under the control of the owner would be privileged relative to less direct forms of ownership. It is less expensive, and therefore more profitable, to defend one hundred small properties owned and defended by one hundred men, than to defend one large property owned, but not defended, by one man.
If the issue is a private good law, a minority in anarcho capitalist society who want it to be illegal would usually be sufficient to make it illegal to do it against members of that minority. If the issue is a public good law, a minority in anarcho capitalist society who want it to be legal would usually be sufficient to make legal for members of that minority to do it.
Let us suppose that X is a private good law: This means that if you are doing X, there is a specific identifiable victim who is likely to get upset with the particular offender, that if you are doing X, you are doing X to someone. That someone may well attack you for doing X. The question then is, to what extent can he get away with attacking people who do X to him, to what extent will he receive backup from his protection agency?
If, however, X is a public good law, then even if lots of protection agencies think it illegal, no one is going to be highly motivated to do anything about it, so if you sign up with a protection agency that is happy with X, then no one is highly motivated to stop you from doing it, and no one is responsible for doing anything about it.
For example, suppose that X is impugning the honor of a man's wife or mother: Calling someone a bastard is held to be legitimate grounds for violence by “God's Soldiers”, but not by most other protection agencies.
Now lets us instead suppose that X is a public good law: For example, forbidding people to smoke dope. There is no particular victim demanding specific action about a particular act of dope smoking. Suppose that most protection agencies believe that dope smoking should be suppressed, but “the Old Hippy Militia” thinks otherwise. Well then, if you are a member of “the Old Hippy Militia” and smoke dope, who is going to interfere? Where is the profit in interfering? You might find it illegal to smoke in some venues - perhaps mall security will arrest you if you smoke dope in the mall. So smoke somewhere else.
In the dope smoking case, there is no profit in going after dope smokers signed up with other agencies, but considerable hazard in doing so, whereas in the calling someone a bastard case, there are irate customers involved.
The practical effect of this is that if a minority want a public good law repealed, it is effectively repealed, and if a majority want a private good law repealed, while a significant minority want it enforced, it is effectively enforced.
In an anarcho capitalist society, you could probably smoke dope, but impugning someone's wife or mother would probably be hazardous, even if a large majority felt that dope smoking was legitimate grounds for violence, and only a small minority felt that impugning someone's wife or mother was legitimate grounds for violence.
Suppose the majority view on calling people's names, and on female chastity is what it is today in our society, but “God's Soldiers” holds the majority early nineteenth century view on these topics: that women should be virgins at marriage, that female adultery should be severely punished, that wives and mothers should have respect and privilege, but not liberty, that a wife or mother gets put on a pedestal and not allowed to get off it, that a man must uphold the honor of his wife for life – cannot openly substitute a mistress, nor marry someone else, formally or de facto, nor withhold support for his wife and her household, except for clearly proven grave fault. It holds that if Bob has sex with Ted's wife, this is legitimate grounds for divorce with Ted retaining all property and children and having absolutely no support obligation, and also entirely reasonable grounds for Ted killing Bob. It holds that if Bob impugns the honor of Ted's wife or mother, this is legitimate grounds for violence, and legitimate grounds for demanding a duel with lethal weapons.
Then I think that clients of other protection services would be most unlikely to call a client of “God's Soldiers” a bastard just as a white man is unlikely to call a black man a nigger in today's society, even if the theoretical backing of their protection agency theoretically reduced the risk of a thrashing by “God's Soldiers”.
Agencies want to protect their customers, but they do not want conflict, and so are apt to enjoin their customers from behaving in ways that lead to conflict, even if they theoretically favor the customer's right to do so. The following notice might appear in the sign up agreement for people signing up with the “Old Hippy Militia”
“In the event that you are killed or injured by a husband while you are impregnating that husbands wife: the Militia undertakes to avenge you, (after due process) unless the marriage contract was registered with one of the following contract enforcement agencies […], or the husband is a member or client of one of the following militias or defence agencies […]”
And the following notice might appear in the “God's soldiers” notification:
“In the event that you are killed or injured by a husband while you are impregnating that husbands wife: may God have mercy on your soul. Your right to protection shall be canceled from the date of the adultery.”
And if “God's Soldiers” succeeded in getting enough defense organizations to agree to tolerate their practices, then a rich man marrying a young wife might well prefer to have his marital contract registered with “God's soldiers” even though he has all his other contracts registered with “Dun and Bradstreet”. On the other hand, perhaps he might not. Possibly his wife might not.
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