Anarchists hold to many principles and codes, and still follow a strict set of personal guidelines and morals. Anarchy is the absence of law, a constitution, a capital, bureaucracy, etc. Not a lack of order. By nature, animals are orderly creatures and if we woke up tomorrow and the government just quit, then we would still have our own order that we would establish within ourselves.
What principles and codes? What personal guidelines? The true purpose of government is to protect the right to freedom. You argue that anarchy would be freedom, but freedom to do what?
If there were no law, people would be free to strip others of their freedoms (kill, rape, pillage, steal, etc.), which is the reason governments were created in the first place.
FOR ANARCHY TO WORK, PEOPLE MUST AGREE TO CERTAIN PRINCIPLES AND CODES. PRINCIPLES AND CODES ARE COMMONLY REFERRED TO AS LAWS. LAWS ARE CREATED BY GENERAL CONSENSOUS (WHICH WE ALSO REFER TO AS THE DEMOCRATIC PROCESS).
If you don't have a collection of individuals agreeing on principles that prohibit the infringement of basic rights (ie. life, liberty, and property) then the "natural order" you speak of will revert to evolutionary principles of dominance.
You said that animals have a natural order, and you're right. It revolves around survival. Survival is a product of competition. In nature, competition almost always creates A dominant figure/faction. Dominance is a form of leadership. When left to its own ends, nature will establish a leader.
Anarchy cannot exist.
You may define it any way you would like, but if you have principles, somebody must establish and protect them. Since they cannot be commonplace, only the individual can protect them. Assuming there is ONE member of a population who's principles are to not regard another's principles as valid, one will eventually dominate/destroy another.
If you define anarchy as having no "common principles", an anarchist society would never be able to agree to establish itself.
Please tell me what point I'm missing, because this seems extremely simple to me.