It seems to be the Uchiha plus Senju bloodlines fused into one that allows for one to gain the power of the Six Paths.
However, recent chapters have suggested that it is really the incarnations of Asura and Indra that allow for this to happen.
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This may indicate that it may not really be just any Uchiha and Senju mixing together into one, or it's likely Obito would have awakened the Rinnegan.
It may actually have to be the incarnations of Asura and Indra. It's not to say that Kagami could just take a piece of Tobirama and awaken the Rinnegan, but it's not really enough info on this to be sure.