There are principles which imply the axiom of choice, and are inconsistent with $V=L$.
For example, the statement "Every well-ordered chain of cardinals between $X$ and $\mathcal P(X)$ is finite" implies the axiom of choice.1 Couple that with "Every finite $n$ is embeddable in the cardinals between $X$ and $\mathcal P(X)$, for some $X$", and you get the axiom of choice, but also the negation of $\sf GCH$. Therefore this principle must imply $V\neq L$.
In this spirit we can in fact generate more than a handful of statements which will imply the axiom of choice, but will violate $V=L$.
In fact we just want an ordinal which bounds these sort of chains globally. Of course, Easton showed that the axiom of choice is consistent with the statement that there is no such ordinal; so this sort of statement is in fact stronger than the axiom of choice.
Truss, John. "On certain arbitrarily long sequences of cardinals." Z. Math. Logik Grundlagen Math. 19 (1973), 209–210. MR317941.
