I think if you read some anarchist philosophy you might be surprised at what you find.

Property can be owned in common. Then it's not private property whose ownership rights are enforced by a State. We have that here in Taiwan with indigenous people and it causes issues with the bureaucracy all the time, which is desperate for a name to put down as landowner. Many societies throughout history have common ownership aka no private property.

Before you ridicule the idea of anarchy perhaps take a look at history - humans as a social animal tend to form societies, not states.