I've seen newspapers handle this[1] by having other phrases for the same referent -- instead of just switching between "the pope" and "he", they'll slip in "the Bishop of Rome" or "the vicar of St. Peter".

Unfortunately, I consider that itself annoying because it then creates the problem of assuming your reader knows which of those are just alternate phrases vs other entities in the story!

Side note: this story got reposted twice recently so my comment on one of them (with examples of me producing a better explanation than the ones I had to wade through) got neglected: https://news.ycombinator.com/item?id=28241558

[1] Oops, another anti-pattern -- saying "this" or "that" when it's not clear what you mean. Here I should have said "this problem" -- it's a great technique for clarifying your own thoughts as well!