What proved that Midway was the location the Japanese were calling AF?

Question

In Midway (2019), the Japanese were using encrypted traffic which the US was codebreaking. The Japanese were planning an attack on a location they were referring to as AF.

Washington believed that AF was located in the South Pacific.

But Layton denies it:

Layton: I heard that Midway accidentally sent out an unencrypted transmission that their water plant was broken.

Nimitz: And is their water plant broken?

Layton: Not that I know of, sir. But that proves Midway is AF.

What exactly proved that Midway was the location the Japanese were calling AF?

Answer 1

As detailed on the Wikipedia page:

After meeting with Rochefort, Nimitz instructs the team to find a way to definitively prove that "AF” is Midway. After Layton instructs Midway to telegraph in the clear (unencrypted) that they are suffering a water shortage, cryptologists working for Rochefort intercept Japanese communications concerning water shortages on “AF”, confirming that "AF" is indeed Midway.

Answer 2

The message about the water system was a plant. The US code breakers devised a false but unambiguous message to be sent in clear from Midway, in the hopes that it would be intercepted by the Japanese radio-listening service and then sent by the Japanese to Tokyo as part of the intel on Midway. Midway does not have a significant natural fresh water source and needs a desalination system to produce said drinkable water.

The plan worked perfectly well. The US first sent their instructions to Midway in a strongly encrypted message unreadable by the Japanese. Midway station followed the instruction and broadcasted the message about the water system in clear. The US listening station then picked up a message (which they could decrypt) on the Japanese network to the effect that point AF had fresh water issues.

Since no other US base had reported such an issue, this proved that AF was Midway.