No less than eleven of his plays were produced during the next ten years. These include the great tragedies Othello, Macbeth and King Lear. His last work was The Tempest, but he may have shared in the writing of the historical play King Henry VIII.
My sense is still that the word 'produced' is focused on writing (and completing the writing of) plays. To me, the word 'produced' is used to talk about Shakespeare's output of written material.
I suppose you could also do some historical research to find out exactly when the plays mentioned were written. However, the dates seem to be about right and the sentence (context) that Hela added in her second post only served to strengthen my impression that the focus was on the writing (output) of plays. Using 'produced' to mean 'staged' or 'performed' also sounds "too Hollywood" to me in