In video filmed Tuesday evening at the palace, the Queen -- joined by Prince Charles and Camilla, the Duchess of Cornwall -- is seen shaking hands with Trump and his wife Melania.
The 93-year-old monarch then looks across the room, apparently to check which world leader she is due to receive next, before noticing her daughter standing in the background by the door.
Queen Elizabeth says something inaudible towards Anne, who can be heard replying: "It's just me," before adding: "and this lot," gesturing towards members of the royal household behind her. The group around the Queen giggles.
It wasn't long before "Princess Anne" was the top trend on Twitter on Tuesday. Some social media users suggested that the Queen was "scolding" Anne for not greeting the president. But according to the PA Media news agency, Anne was standing to one side because she did not form part of the official welcoming party.
The Queen was apparently looking up to see who was due to be greeted next, PA reported, and Anne's joke referred to the fact that Trump was the last of the NATO leaders to be presented to the monarch.
That video appeared to show British Prime Minister Boris Johnson, French President Emmanuel Macron, Canadian Prime Minister Justin Trudeau, Dutch Prime Minister Mark Rutte and the princess laughing about the US President's behavior during the summit.
When Trump was asked about the exchange later, he called Trudeau "two-faced," before adding, "honestly with Trudeau, he's a nice guy."
Microphones could only pick up snippets of the conversation at the reception, to which the press was given limited access. None of the leaders seemed to be aware that the conversation was being recorded. They were, however, talking openly and loudly enough to be heard by others.
Correction: The headline on this story has been corrected to better reflect the circumstances of the Queen's gesture.