Imagine a language with a gender system along the lines of German. Now, however, imagine that the society has gone through a quick but fairly successful modernization, where a formerly very strict division into 'female occupations' and 'male occupations' has over a generation or two become rather 'ideally equal' for some combination of those words.
Ok, look, I am trying very hard not to take any stance in that debate. This is all set-up for a convoluted grammatical situation.
However, the titles associated with occupations persisted in the gender they previously had been associated with. So ...
Sergeant is still masculine when it is a woman sergeant.
Secretary is still feminine when it is a male secretary.
Some limited examples of this can be found in Europe to this day, with some titles in French, for instance, only having masculine forms, and in some varieties of Swedish, sjuksköterska, "nurse", only having a grammatically feminine form.
However, the twist I am going for is one where pronominal binding still is lexically gender-based even when the gender of the particular person is known. When referring to Tim the secretary or Jenny the sergeant, the gender of the pronoun would follow the gender associated with the occupation.
However, when speaking of Tim or Jenny as persons having private lives and so on, they would get their expected pronouns.
This creates a situation where persons working in occupations associated with the other gender can get their professional person and their private person separated by pronouns, but people working in gender-typical occupations do not have this quirk available to them.