Knaster-Tarski's theorem States that if $(A,\le)$ is a complete lattice, then every monotone function $f :A \to A$ has a fixed point. The proof is carried out in $\mathsf{ZF}$.
By $\mathsf{KTC}$ we mean the statement that the converse holds, that is, if $(A,\le)$ is an incomplete lattice, then there is a monotone function $f : A \to A$ without fixed points.
It turns out that this was proved by A.Davis here. However, the proof uses (at least apparently) full choice. I Would mention that the part that uses the axiom of choice can be reduced to prove that if $(A,\le)$ is a lattice with no top element, then it has an unbounded subset wellordered by $\le$.
The question then is ¿How strong of a choice principle is $\mathsf{KTC}$?
I suspect it does not imply $\mathsf{AC}$ over $\mathsf{ZF}$, however I haven't been able to prove the principle with anything weaker.
In the other direction I have been able to prove only weak principles over $\mathsf{ZF}+\mathsf{KTC}$, namely that for any infinite $X$, $\mathcal{P}_{fin}(X)$ is D(dekind)-infinite: It is clear that for an infinite set $X$, $\mathcal{P}_{fin}(X)$ is a noncomplete lattice, and thus by $\mathsf{KTC}$ has a monotone function with no fixed points $f : \mathcal{P}_{fin}(X) \to \mathcal{P}_{fin}(X)$. Then is clear that $\varnothing \subseteq f(\varnothing) \subseteq f(f(\varnothing)) \cdots$ is an increasing sequence, and is strict since $f$ has no fixed points. Thus one can inject $\omega$ in $\mathcal{P}_{fin}(X)$.
From this one can deduce that in $\mathsf{ZF}+\mathsf{KTC}$ the following hold:
- $X$ is finite iff $\mathcal{P}(X)$ is D-finite, whereas in $\mathsf{ZF}$ one only gets $X$ finite iff $\mathcal{P}(\mathcal{P}(X))$ is D-finite.
- Finite $=$ D-finite for linearly orderable sets: Let $(X,\le)$ be a linearly ordered set, and take an strictly increasing function $f : \omega \to \mathcal{P}_{fin}(X)$ then $g : \omega \to X$ given by $g(n) := \min(f(n+1) \setminus f(n))$ is injective.
