There is no such thing as a "complete theory of arithmetic" - see Godel's incompleteness theorems.

The above statement isn't true. What Godel's theorem says is that any effectively generatable theory which is strong enough to model PA is either incomplete or inconsistent. Effectively generatable for our purposes can be taken to mean that valid proofs in the system are enumerable and that there's a primitive recursive procedure to determine whether a given proof is valid.

If one removes the effectively generatable condition then one can easily produce a complete, consistent system. Consider for example the following system: Consider some lexicographic ordering of well-formed statements in the language of PA, and let S(i) be the ith statement in this list. Let P(0) be PA, and generate P(n) by asking whether S(n) or ~S(n) is a theorem in P(n-1). If so, then set P(n) = P(n-1). If not, then let P(n) be P(n-1) with the additional axiom of S(n). Now, consider P(infinity) defined as the union of all P(n). This theory is complete and consistent but it is very much not effectively generatable since we can't even recursively enumerate the system's axioms.

It might help for you to take a model theory or logic course since there are a lot of subtle issues here that you seem to be missing.