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Reiko wants to be the emperor one day. He has no real attachment to Mileena, only wanting to keep her happy and further his goal of taking the throne.
If he had a significant other, he would keep them a secret. He wouldn't want Mileena or anyone else to find out.
Choosing between the throne and his significant other isn't a choice that he's going to make. He hates the idea of even having to make that choice. If he had to, it would depend on the situation and circumstances that led to this problem.
His significant other would reside in the same place that he does: the castle.
He isn't one for PDA; all of that would be reserved for your private quarters. He has a reputation to uphold and wouldn't want anyone to think that he's weak or use you against him.