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I don't think that Yui's a natural hardass. I don't think that's her real personality. Her real personality is more like the girl we saw in the first two episodes of this anime, just more mature.
Deep down inside, Yui is a nice person. But she's a fierce patriot. Yuuya set off her patriotism/nationalism, just like she set off his dislike for the Japanese side of his ancestry. That caused Yui to act like a complete hardass towards Yuuya, as he offends her patriotic pride.
But once Yuuya and Yui are put in a situation where nationalism isn't that important any more, their real selves come out better.
Also, I think that Yuuya's push-back has actually impressed Yui some. The guy doesn't take the brow-beating without fighting back as much as the rank differences will allow (and arguably then some). At some level, Yui probably respects Yuuya for sticking to his guns and not backing down from his side of things, as much as she may disagree with him.
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