Death, where is your sting? Hades, where is your victory? (1 Cor 15:55)
For natural law theories, property rights are a formidable problem, because our animal nature is fully compatible with theft. There can be made an argument that property is needed for one's survival, but this does not involve all property or the survival of the others that may depend on alienating one's property. So one needs to appeal to some other nature than biological one; e.g., one's rational nature.
One school of natural theology is that this "other" nature is the better nature we were endowed with during the Creation; the Fall corrupted this original nature but it still has a hold on us, and the moral norms are the remnants of this nature that is cast on our hearts. This nature is not what we presently are but what we supposed to be, ought to be, and will become. We've started as sin-free immortal creatures; it is our own fault that we are mortal, sinful, and thieving. On this theory, property rights derive from our prelapsarian nature, and to explain property one needs to explain how the protection of property naturally emerges from this nature.
We do not have too many examples of immortal animals around, but we do have some polyps that can endlessly regenerate themselves, and T. nutricula mentioned in the previous post is the most complex example of this kind: after releasing the gametes, its adult medusa form undergoes transformation back to the juvenile form, the colonial polyp state, by cell transdifferentiation. Like most other animals, we gradually age and die through senescence after we reach sexual maturity and produce offspring. In contrast, these jellies go back to the cradle, so to speak. It is the endless cycle of maturation followed by rejuvenation, back and forth, back and forth. This cycling is the most advanced form of immortality seen on this planet.
Suppose you have immortal rational creatures of this type. I argue that the sanctity of property is the natural condition of such creatures. Indeed, as the regeneration occurs, all memories of one's self disappear: the new self is the same genetically, so the death is cheated, but the personhood does not survive, there is no continuity of experience. The old self and the new self cannot communicate biologically, so the only way of preserving continuing personality is through externalizing it: creating a reflection of one self in the material world that serves as a template for regaining one's older self after the regeneration. This material reflection is one's property. You can think of it, say, as a diary that your older self writes for the younger self guiding this younger self towards its previous incarnation. But it does not have to be a manual - anything would do to provide the connection. It serves another role, too: as the memories are being lost during the rejuvenation (which may take as long as our aging), one needs the reminders of one's own past; again, think of keeping mementos reminding you of your own past, but subtract the morbidity.
It is easy to see why rational creature of this type would not destroy or steal property. Property is the repository of personality and stealing/destroying property means destroying this personality: the personhood is preserved by means of all these connections between the self and the material world and it is required for continuity of personal existence, however imperfect. Theft means murder, it is just another way of achieving the same end. You shall not steal is another form of "you shall not murder" in the world of such creatures. Transfer of property is the transfer of one's personality to another being: an act of ultimate self-sacrifice.
We are the inverse of these creatures: from their middle life they become younger and get reborn and we become older and die. These type of creatures ARE our own distant eumetazoan past; their nature is the very "elusive" higher nature the natural theory seeks in the prelapsarian man. This nature was our natural condition and - who knows? - it can be our future condition again: the cellular toolkit needed for continuous transdifferentiation does exist for us to use; we've been provided. I think that property still serves the same role for us as them, although this role has been corrupted, as all other parts of our nature: we externalize our personhood to communicate our younger selves to our older selves and to our children. Retaining property is less crucial for us, but the sacred status of property could be the gold standard for rational creatures in general; quite possibly, such a life cycle is the only possibility of achieving physical immortality compatible with our animal nature.
So there is a nature that makes property rights natural, in the most unambiguous sense.