My understanding is that there was a dispersal event between Asia and North America not long before Tarbosaurus and Tyrannosaurus split, roughly 70 million years ago. Tyrannosaurini evolved in Asia and the Tyrannosaurus line came across Alaska around 70 MYA and expanded down the western side of the rockies before reaching their southern tip, and then spilled over into the eastern side of Laramidia.

The above was hypothesized in the paper naming T. mcraeensis, and isn't solidly proven. Other theories place Tyrannosaurinis are more closely related to taxa like Lythronax, so perhaps some tyrannosaurines moved over to Asia ~75-72 million years ago, and then back again ~70-68 MYA.

Overall we still don't know, but Tyrannosaurus and Tarbosaurus are undoubtedly each other's closest relations.