For that you need an arbitrator that's recognized by both parties because perfect arbitrators don't exist and in the absence of government there's no court with unquestionable authority.
It is true that there are no perfect arbitrators. However, there may well be courts with sufficiently-broadly-accepted authority.
For example if I was some ruthless libertarian billionaire living in a highrise surrounded by slums one of the people from the slums could sue me for some triviality (like the shadow of my highrise depriving the slums of sunlight) and suggest a leftist court for arbitration, a court that is overwhelmingly favored by the local population, but one that I, the billionaire, cannot agree to because I know the court favors the people from the slums and would like nothing better than to make me pay huge damages to the people from the slums (like a government taxing the rich to take care of the poor).
By the same token, if you are a ruthless statist billionaire living in a highrise surrounded by slums, a majority of slum dwellers could elect a city council that would impose an outrageous property tax on you, or even set a death penalty for casting shadows.
Clearly, you would appeal to your state court and, if necessary, the USSC.
The same options are open to you in an an-cap.
Let me take a step back and clarify. For an an-cap to be a peaceful a stable society, the exact same precondition is required as for any other society, namely that there is very broad public consensus over what constitutes legitimate means for settling disputes.
In the US today you will find very divergent ideological views, from red-neck neo-con fundamentalists to bleeding-heart liberal socialist atheists.
They cannot agree on whether abortion should be legal or not. They cannot agree on whether the US Constitution implies abortions should be always legal, never legal, or sometimes legal.
But they generally (with very few exceptions) do agree that rulings of the USSC are legitimately binding, even when they don't like those rulings.
A similar convergence must and will take place in an an-cap society. There won't be a single body like the USSC, but rather a small group of highly reputable super-arbitrators, together with a larger number of normal arbitrators, all of whose legitimacy is ever conditional on the public perception that they are ruling in accordance with the fundamental law of the land, the NAP.
Any other organisation (e.g. enforcement agencies) would similarly be expected, by broadly-shared public opinion, to comply with the NAP as "interpreted" by those arbitrators.
This is comparable to broad expectation in today's America that all agencies, whether public (e.g. police forces) or private will comply with the US Constitution as interpreted, ultimately, by the USSC.
Further, certain norms would be expected with respect to the criminal justice process, namely that responsible parties (i.e. anybody not considered an outright outlaw) would reasonably cooperate with the process. It is OK to object to a particular arbitrator, as long as you are reasonable about the choice of another.
Jurisdictional disputes (very common in today's society) may well arise, when two courts reach contradictory conclusions in a particular case. But just as in today's society, courts have a way of avoiding such disputes, with rules evolving to resolve them.
In Israel, for example, two separate sets of courts have jurisdictional over family matters - "civil" family courts and Rabbinical courts. Over time, the legal rule that evolved is "first to file" whereby in whichever court a claim was first filed has jurisdiction, which the other court respects.
In today's globalized economy there are countless cross-border disputes, just as there are countless cross-state disputes within the US. All of them are resolved peacefully every day.
So while the preferred scenario is indeed one in which both sides to a dispute reach an agreement as to the arbitrator to resolve their differences, the system cannot be crippled by one side refusing to cooperate.
Compare to the current criminal justice system. It is better if the defendant cooperates with his lawyer in his own defence. However, a defendant which refuses to speak at all is still going to be sentenced if found guilty.
If public opinion forces a particular court on me because they trust the court (or they just like the most likely verdict this court will issue) this is no different from an elected government forcing me to do this.
You are focusing on the procedural rather than the substantive aspects of the action.
You can reasonably argue that such process is akin to government justice in that it is applied onto you against your will.
But substantively, the key difference is that while government action is routinely aggressive, the system here described comes as close as humanly possible (i.e. errors are always possible) to avoiding any aggressive action. Whether you agree with that or not.
So the system might look to you like government in that one respect, but in fact it isn't because force would only be authorised when it is deemed defensive.
To avoid a show trial I have to have faith in the objectivity of the court that arbitrates our dispute.
No. But society at large needs to have that faith. If you are being unreasonable and reject a court that society finds acceptable, you are on your own. Often literally.
You have to prove beyond reasonable doubt that I stole your tv and then it really does matter which court you're appealing to (they decide what's reasonable doubt) because there is no perfect court on this Earth and there never will be one either.
I'm not sure "beyond reasonable doubt" is the appropriate standard. In civil cases today, for example, the standard is "preponderance of evidence".
Free men are not equal and equal men are not free.
Government is not the solution. Government is the problem.