Could a universe with a relativity that is a cross between Galilean and Lorentzian Relativity be self consistent?

I was thinking of a universe in which there is a relativity that is between Galilean and Lorentzian Relativity. In this universe there would be effects of length contraction and time dilation but it would not be according to the Lorentz Factor. In this universe there would be no cosmic speed limit but the faster an object would move the harder it would be to accelerate. There would be no special reference frames and the laws of physics would be the same in all reference frames moving with constant velocity.

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Could a universe with a relativity that is a cross between Galilean and Lorentzian Relativity be self consistent?

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