It is well-known that $2^{\aleph_0} \nrightarrow (\aleph_1, \aleph_1)^2$. The usual proof uses a well-ordering of $\mathbb R$ to find a colouring depending on the difference between the usual order and well-order of $\mathbb R$, and so it uses the axiom of choice.
My question is: Can we use the statement $2^{\aleph_0} \nrightarrow (\aleph_1, \aleph_1)^2$ to well-order $\mathbb R$ in ZF? If not, then what choice principles can we prove in ZF + "$2^{\aleph_0} \nrightarrow (\aleph_1, \aleph_1)^2$"? Thank you.