I would say they are mutually exclusive. It's not about the question itself, but the restriction that the tags put on the answers and the conflict it creates. As a summary of the tags
- Hard Science: answers backed up by equations, empirical evidence, scientific papers, other citations, etc
- Reality Check: Asks if a given concept is realistic in a given context. Answers should say yes or no, with supporting info
Now let's just ignore any magic based questions. They will all automatically fail the [Hard-Science] tag because well.. there isn't exactly Hard Science behind magic...
So looking at some of the more popular question with both tags
Stealth in Space: How realistic is it?
Under the Hard-Science tag, the answer is No, you can't stealth. You have no way to prevent the detection of certain aspects of the spaceship.
Under a reality check tag, the answer would be Yes. It's plausible and present in other worlds, e.g. Star Trek (I'm sure there's more since stealth tech is a huge part of some games )
Could humans alter the moon's orbit significantly with current technology?
- Under the Hard-Science tag, the answer is again No. Way too much energy
- Under a Reality Check tag, the answer would be Yes, it's plausible. Again I'm pointing to commercial worlds like Despicable Me or Bruce Almighty.
How quickly can I form a mountain chain?
- Under the Hard-Science Tag, the answer is Yes
- Under the Reality Check Tag, the answer is Yes
In Each of these questions, the Hard-Science Tag quickly overrides the Reality Check Tag. Is a concept realistic? Sure, but then you need to back your answer with equations, evidence or papers, so instead of a quick reality check, your now creating a Hard-Science answer because of the much stricter conditions the tag places on answers.
So any answer that are [Hard-Science] are also [Reality-Check], but not all [Reality-Check] answers are [Hard-Science]. In fact, if you want to combine different parts of [Hard-Science] it makes more sense to go with [Science-Based] because you could apply two conflicting scientific principles and it would still be valid answer under [Science-Based] but not under [Hard-Science].
So in summary: [Hard-Science] restrictions override [Reality-Check] restrictions so answers have to confirm to the [Hard-Science] tag making the [Reality-Check] tag redundant.