A Times dispatch about the presidential campaign of Ben Carson refers to him as "a retired pediatric neurosurgeon" but also, repeatedly, as "Mr. Carson" rather than "Dr. Carson." On Twitter, a Times social media editor responded to my question about it by citing Times style to the effect that "Dr. should be used in all references for physicians...whose practice is their primary current occupation."
That does indeed seem to be Times practice for Republican physician-politicians such as Ron Paul and Tom Coburn, who get the "Mr." treatment from the Times. Yet when it was a Democratic Vermont governor-doctor running for president full time, it was "Dr. Dean" on second reference in Times articles such as this one, this one, and this one. And when a Democratic secretary of state with a Ph.D wanted to be "Dr. Albright," the Times let her have her way.
The Times social media editor suggested that it "might make for a good public editor post if we're being inconsistent." Margaret Sullivan, call your office.