I now follow a new writing rule that solves the gender bias problem while making all pronouns agree with their antecedents.
Feminist demands have caused a major conflict in prose, such as in the following sentence.
Every radiologist worth his or her salt knows that!
As a result, writers—even good writers—have begun to incorrectly write:
“Every radiologist worth their salt knows that!”
That is so very, very bad.
From now on, female writers write the feminine pronoun:
Every radiologist worth her salt knows that!
And male writers write the masculine:
Every radiologist worth his salt knows that!
The reader will quickly acclimate to this change, realizing the writer is writing according to his or her own gender. This solves the awkward antecedent problem without offending sensibilities.