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Eliezer_Yudkowsky comments on Particles break light-speed limit? - Less Wrong
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That might preserve before-and-after. It wouldn't preserve the locality of causality. Once you throw away c, you might need to take the entire frame of the universe into account when calculating the temporal successor at any given point, rather than just the immediate spatial neighborhood.
There could be some other special velocity than c. Like, imagine there's some special reference frame in which you can send superluminal signals at exactly 2.71828 c in any direction. In other reference frames, this special velocity depends on which direction you send the signal. Lorentz invariance is broken. But the only implication for local causality is that you need to make your bubble 2.71828 times bigger.