If $X$ is a Tychonoff space, then using the Tychonoff theorem and thus the full axiom of choice, it follows that $X$ admits a Hausdorff compactification.

My question is : what remains true if we do not have the axiom of choice, but still allow weaker versions like Dependent choice (DC) or countable choice ?

More concretely, would this result still be true under (DC) if we add a stronger condition on $X$ like being metrizable, or second-countable ? In those cases, could we obtain a metrizable compactification ?

thinkDC should suffice for the second part (but I'm not sure). $\endgroup$ – Will Brian Dec 6 '18 at 15:21