I read this short story in an old Science Fiction anthology I checked out from a library. It's bugging me that I can't seem to recall the author or title.
The story is narrated by a Native American person, possibly a wise-man or shaman of some sort, as if relating the story to someone, with the occasional excerpt from Shakespeare's diary.
The plot was that Shakespeare, back when he was a no-name playwright, had ended up stowing away on a ship bound to America. Somehow he ends up with the natives, where he manages to communicate his name by shaking a spear. He manages to teach the natives English, where the narrator remarks something to the effect of how it's weird that English isn't tonal. The narrator also expresses a superstitious aversion to having his name written down, and has a dismissive attitude towards the concept of writing things down at all.
Despite this attitude, Shakespeare manages to write and produce Hamlet. To his chagrin he ends up having to recruit women to act the female parts. When the play premiers, the natives regard this play as a comedy.
I remember there was an author's afterword about the historical basis for this, ending with a comment addressed to people who think Shakespeare's plays were written by (I believe the wording was) "the Earl of Stratford, Queen Elizabeth or Elvis Presley": "Hah, hah and once again hah!"
I think the anthology also contained the story "Even the Queen", where medicine has learned how to stop the menstrual cycle, and there's a movement called "cyclists" that feel this is a tool of patriarchal oppression. Then there was "Wang's Carpets", where the titular otherwise uninteresting sea creatures turn out to be molecular computers simulating life in multi-dimensional frequency space. Yet another possibly included story had people sending out clones of themselves to explore the galaxy so they could absorb the clones' memories in a sort of tourism-by-proxy setup.