Secretary of State Rex Tillerson’s frustrations with the White House have been building for months. Last Friday, they exploded. The normally laconic Texan unloaded on Johnny DeStefano, the head of the presidential personnel office, for torpedoing proposed nominees to senior State Department posts and for questioning his judgment. Tillerson also complained that the White House was leaking damaging information about him to the news media, according to a person familiar with the meeting. Above all, he made clear that he did not want DeStefano’s office to “have any role in staffing” and “expressed frustration that anybody would know better” than...