Basically, when the communists won in the 1940s, the ex Tsinghua principal escaped to Taiwan with some of his teachers and board members, and they founded Taiwan Tsinghua. In a while, both claim to be the "rightful" Tsinghua, but they have resolved now and agreed that both are the rightful owner of the history of the prewar Tsinghua. But if you talk about the current situation, they are just two separate schools with a more than usual close relationship.
