I've often encountered topological spaces like $2^\omega$, where there is a short proof of compactness using Tychonoff's theorem and a long proof of compactness not using Tychonoff's theorem. Are there situations where this is not the case, i.e. the compactness of the space depends on Tychonoff's theorem?
Essentially, I'm looking for any examples of topological spaces $(X,\tau)$ such that
1. $(X,\tau)$ exists in ZFC.
2. $(X,\tau)$, or something more or less equivalent, exists in some model of ZF without choice.
3. $(X,\tau)$ is compact in ZFC but not in the other model.
By "more or less equivalent" I mean that they are constructed in the same way, or have similar definitions, or have the same name, or anything else that would make them analogous; my knowledge of set theory is not particularly deep, so the notion is a little fuzzy.