I thought AC is equivalent to saying that every surjection has a section

surjections in each direction implying bijection is weaker, but sections of all surjections does imply AC

@TobiasKildetoft Actually, it's open if it's weaker. But we (read: I) expect it to be weaker. There's also "If there is a surjection, then there is an injection in the other direction [but not necessarily a section]" which is stronger than what you mention, but also it is open about whether or not it is weaker than choice. Again, I expect the answer to be that it is weaker.