Each person may designate one person or organization as their "law agent". If A has a dispute with B, then a decision on the case is made by the prior agreed-on arbitrator between between A and B or their law agents, if one exists. These agents might have a direct agreement specifying their mutual arbitrator, or might belong to larger coalitions which specify default arbitrators regarding disputes with other coalitions (or might even post contract-parts, which be joined into a whole contract my matching with other posted contract-parts).
Arbitrator rulings are "final" in the sense that a dispute between a party and her arbitrator is handled just like any other dispute. Arbitrator rulings are enforced only by reputation and shunning; there should be an equilibrium where people rationally avoid dealing with those who don't abide by arbitrator rulings. If two people don't have an arbitrator, then there is no law between them.
Finally, the "contracts" specifying law-agents and arbitrators should be simple and literal, with very narrow grounds for non-enforcement. Pretty much the only thing everyone needs to agree on is how to tell who is someone's law agent or arbitrator, and how strongly to shun someone who doesn't abide by a ruling, or who doesn't shun such a person.