After the defeat of France by Nazi forces in 1940, an armistice was signed between Germany and France. Paul Reynauld resigned as Prime Minister of the French government and was replaced by the aged Marshall Philippe Petain (who was even older than Churchill.) Petain was a popular figure and hero of the Great War-he had single handedly prevented the collapse of the French army in 1916 when it looked like the German onslaught would finally succeed and was perceived as being quite charismatic and a bit of a womaniser. He was an advocate of ending the war and seeking a peace settlement with the Germans. He was voted in as PM by the French cabinet and immediately sought and secured an armistice and surrendered to the Germans. As part of this peace settlement the Germans occupied only certain parts of Northern France, Paris and certain areas of the West coast (in order to gain access to the ports to base submarines there to harry UK shipping). There was also a fairly large ‘free zone’ in the South, which was not occupied by the Germans and was headed by Petain and Pierre Laval based in the spa town of Vichy-Vichy France.
Vichy France was, in theory, an independent country. It’s government could, in theory, decide its own policies on any matter. In reality, it was merely a vassal state of the German Reich that more or less automatically aligned its policies with those of Germany anyway.
This is the situation that would pertain under Theresa May’s Chequers Plan. It would turn the UK into a version of Vichy France- a vassal state of the EU that is technically independent of the bloc but in reality is informally subservient to it and that has its policies on many issues decided in Brussels with Theresa ‘Marshall Petain’ May complying with her master’s wishes by aligning the UK’s policies to those of the EU anyway automatically.