Once again, one must ask the question what do you mean by "the full product" of labor? Remember, labor is the application of brain and/or muscle power to something to create something else of new value. But what was the original substance that you applied your brain and muscle power to? Most likely the raw materials were not "raw" in the sense that you personally gathered those materials from nature.
But even if you had personally gathered those raw materials, that was only possible because you are able to wear clothes and shoes, figure out a way of transporting them back, and then building whatever it was from the raw materials that you had gathered.
Does the fact that all labor is social somehow negate property rights, though? If the chain of custody is legitimate, then why is it problematic? I know how to make rope. If I go to the flax farm next door, and agree to help the farmer harvest the crop in exchange for a few sheaves of flax, then is the rope I make somehow not 'mine'? He got his seeds and tools by voluntary, legitimate means, too. (Let's assume we're in an anarchist collective and not capitalism, where objects really are produced illegitimately via organized theft of surplus labor.)
Personally I agree with the anarchist idea of 'usufruct rights' to possessions posted from the Anarchist FAQ above. I don't think it comes from not being able say exactly who made something, though. There has to be another principle at work, and I think the difference between possessions and capital holds the key. You can use possessions yourself, but capital requires a social relationship. If that social relationship is coercive, then the property is theft.
"This is my test of character. There you have the despotic instinct of men. They do not like the cat because the cat is free, and will never consent to become a slave. He will do nothing to your order, as the other animals do." — Jean-Jacques Rousseau.
"The intellectual and emotional refusal 'to go along' appears neurotic and impotent." — Herbert Marcuse.
"Our blight is ideologies — they are the long-expected Antichrist!" — Carl Gustav Jung