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Economic relationships didn't stop WW1. I mean it's always a good idea to have another layer of protection in form of financial losses on part of the assailant but if it's the only card they have that ancap territory (can ancaps even form countries?) could simply get subjugated in one way or another and provide the service at terms primarily beneficial to the stronger partner, most likely having to surrender some freedoms as well.
There isn't exactly much land that doesn't have some state's claim on it. Getting an entire existing state to go full anarchy is a tad unrealistic. You can convince a local population but seceding will, even if done peacefully, destabilize the area and is going to hard to legitimize. Not that nothing even remotely similar was ever achieved eg. Switzerland but there's enough to suggest they had Templar military and monetary support and those were simpler times.