As an AnCap, I have no direct issue with AnComs so long as they extend the same respect for personal direction as I afford them. Meaning, I will not interfere with an AnCom community if they decide to do create a community centered around their ideas that is voluntary. The problem (and maybe you are different from others) is that I have yet to meet an AnCom willing to leave me and property alone like would be willing to leave them alone.

I think the point at which AnComs get property wrong is in the concept of self-ownership. We own ourselves and control the scarce resource known as our body and if this true then we must be able to own and control other resources outside of the body in the environment we inhabit in order to exercise the self-ownership. 

"The spurious logic of the dialectic is not open to the left-wing anarchists, who wish to abolish the State and capitalism simultaneously. The nearest those anarchists have come to resolving the problem has been to uphold syndicalism as the ideal. In syndicalism, each group of workers and peasants is supposed to own its means of production in common and plan for itself, while cooperating with other collectives and communes. Logical analysis of these schemes would readily show that the whole program is nonsense. Either of two things would occur: one central agency would plan for and direct the various subgroups, or the collectives themselves would be really autonomous. But the crucial question is whether these agencies would be empowered to use force to put their decisions into effect.

All of the left-wing anarchists have agreed that force is necessary against recalcitrants. But then the first possibility means nothing more nor less than Communism, while the second leads to a real chaos of diverse and clashing communisms, that would probably lead finally to some central Communism after a period of social war. Thus, leftwing anarchism must in practice signify either regular Communism or a true chaos of communistic syndics. In both cases, the actual result must be that the State is reestablished under another name. It is the tragic irony of left-wing anarchism that, despite the hopes of its supporters, it is not really anarchism at all. It is either Communism or chaos."
-Murray Rothbard

We can think on the implications of this natural right to one’s self by thinking on the other options if this idea was not true. We have two options beyond self ownership:

One class of individual has the right to own another class of people.
Everyone is entitled to an equal share of everyone else.2

The first scenario specifies that one class of individuals identifies as human while the class that is owned is considered “subhuman”. The owned class is entitled to none of the rights of the owner class because they really are not human. But the idea that the “subhuman” class is not human is incorrect because they really are human. Science has shown that all “people” are human on a genetic level and to say otherwise is to advocate for slavery by another name. Most “reasonable people” (and most people consider themselves “reasonable”) do not advocate for slavery. Therefore we can throw the first alternative to self-ownership away.

The second option, that each individual is entitled to the own an equal share of everyone else, is similarly false and even impractical. This is because this option assumes that society supersedes the individual, thus the collective is entitled to own you. But society holds no special place over the individual since society is itself dependent on the individual for its own existence. This is made clear by the fact that if the component parts (the individuals) disperse, so does society. Therefore, how can something like society that is dependent on the individual for its existence take precedence over that which it is subject to? Secondly, this is inefficient. Ethically, if each person is entitled to a piece of everyone else and therefore a piece of every resource, no individual could act without the consent of every other individual. People would die before consent could be given and this directly contradicts the idea that we can act in order to survive. Therefore we can throw the second alternative out as well. Pragmatically, logically and ethically the second option does not make sense.

Put into economic terms "personal property" would be roughly equivalent to consumer goods, whereas "private property" would be equivalent to capital goods.

Consumer goods are generally the final product which gets produced through a production process. Examples would be food, housing, personal transportation.

Capital goods are generally only used for the production of other goods, either consumer goods or other capital goods. Examples would be tools, factories, raw materials.

The combination of labour with the correct capital goods can produce a much higher quantity and quality of goods (consumer or capital) than labour alone. E.g. fishing with a net catches more fish than just with your hands.

In the capitalist system, you voluntarily entre into a contract in which you exchange a certain amount of labour for a certain amount of money... You will apply that labour to the capitalists means of production which he bought with is own money (which in the end, maybe decades or centuries ago, was made by work). 

Everybody is free to go and buyhis own means of production and becoming self-employed. 

Taking the means of production and putting them in the hands of government is theft, and will eventually not solve anything. Watch Venezuela for the results of such a policy. 

Throughout the ages wage labour has existed, and throughout the ages, selfemployment has existed. Because of the ever larger capital investments which are required to build modern products, wage labour is becoming more and more important. 

Don't like where this is going? Because the capitalists will own everything? That's the beauty of the sysem. Everybody is a capitalist, and everybody has a stake In iy. Whether it be your pension fund, or your private savings, you are undoubtedly invested in companies, you undoubtedly directdly or indirectly own means of production which others work on, while you work on means of production which are owned by others.

"Before capitalists and feudalists got a hold of private property nobody owned it and everyone kept what they themselves produced." -- What time period are you referring to in this statement?