I do not consider myself a Leftist, but...all forms of anarchism historically, from the anarcho-commies to the free market mutualists, have advocated at least the central negative goal of socialism: the abolition of profit, interest, rent, and compulsory wage labor. Socialism originally did not equal statism but rather collective self-organization by the workers. You can have statist and libertarian forms of socialism (supposedly).
"Self-ownership" as libertarians call it does not automatically lead to a right to property because many different notions of legitimate property exist: access by need (communism), access by community (communitarianism), workplace ownership (syndicalism), ownership by homesteading and use (mutualism), ownership by possession (egoism), non-ownership (green?). None of them have any fondness toward putting a fence around food to let it spoil, nor owning humans, nor class-based ownership of capital. We have specific thoughts that qualify "property", most of us differentiating between private property versus personal possessions, commodities versus belongings, that used passively to accumulate wealth and that used actively for subsistence.
Anarchists distrust monopoly, and have critiqued the elements of monopoly inherent to various forms of property (e.g. one group owns all the productive tools of society). You cannot just say anarchists believe in a natural right to property; natural rights are social fictions! Most of us have no solid position in the deontological versus consequentialist justification arguments anyway.
Also, Tea Partiers I've talked to do not believe in "limited government", they believe in small government for the white middle class patriot man, and big government for everyone else (via militarized borders, war on terror, domestic spying, colonialism, war on crime, war on drugs, etc).