The first thing that I did was zoomed out, surely Shakespeare wrote about something in the New World?
No. It is amazing how small his world was. He was born and grew to adulthood, in the world where Spanish dominance kept England from attempting to explore the world. While Jamestown was settled before he died, he never wrote about it.
I've updated my understanding of how (un)aware people were in this era of the larger world. I have no idea why I would have ever expected otherwise.
The map contains a bunch of references to America, the West Indies, Guiana, and Mexico. (Often with a connotation of "faraway exotic place" or "exciting new international development".)
He may not have written about the British colonies but the New World was clearly at least somewhat present in his mind and his audience's minds.
A lot of that is bias from the fact the whole map was vibed/hallucinated by an LLM instead of just sourced from (what I'm sure are many) concordances of Shakespeare's works.
For example, "The Tempest" famously mentions the "Bermoothes" (Bermuda), but that's not included in this LLM's output for some reason. Any decent subject-index of Shakespeare would include it.