We will be in London Oct 1-10 and wanted to see a production the The Globe Theatre. It appears from the website that nothing is available during our stay. Suggestions?
It appears that the Merchant of Venice is indeed sold out during your visit, but there are tickets available for Imogene on October 2, 5, 6, and 8.
It seems like The Merchant of Venice is running at the Globe Theatre during the dates you are in London. https://tickets.shakespearesglobe.com/eventlist.asp?ShoCboEvent=True
Hope you can get tickets, it is a wonderful experience.
EDIT: now I see the Sold Out designation. It would be good to go to the box office as soon as you are in London and ask about ticket returns for any of the dates you'll be in London. It never hurts to ask.
"Imogen" is a retitling of the play normally known as "Cymbeline" after its actual protagonist.