I still don't see how any sort of capitalism or socialism can exist at all in the form we would recognise, when no government exists. Capitalism today may cite the government as its enemy, but it is important to recognise that the system did not evolve up from simple agrarian tribes; rather, it arose out of decentralizing the sort of profit-ventures introduced and undertaken by government mercantlists.
I have already stated that I do not believe in anarchism, of any economic bent. It can lead to nothing more advanced than social-Dawinism, and will provide nothing more-refined than simple necessities like force and sustenance.
With respect to capitalism, I argue that while government intervention may be bad after a point, it still requires government ultimately, because it depends firmly upon the existence of money. Given this, a person who has money will grant another money to help him, by selling him a product, capital, or services. A certain degree of mutualism is sponsored when peoples' greed is displaced to a commodity that they all need to better themelves, and in turn, need to cooperate with others to earn. Without government-produced money the temptation is too great to simply mint one's own money, or steal, or to lapse back into a barbaric sort of greed. Lest you think that money is the root of all evil and nothing-but, I remind you that without it, we would all be going to the market with guns to get what we needed.
That in mind, socialism of a sort can and will exist in a system of anarchy, but it is nothing like the utopian anarchist world Marx envisioned. People will redistribute everything and there will be no necessary permanent view of private property, but the majority of commodities will end up in the hands of the strong, because people are greedy, and it is innevitable that authority will generate out of strength. So while I believe citizens should have the right to own guns, and own businesses, simply giving them all those things is not the way to ensure equality. Somebody's going to always be a better businessman or better shot, so what starts as a system of competition inevitably degenerates into bully-reign.