According to this widely shared Quora answer, the following interesting story took place when the Mongolian president visited the UK in 1996:
In 1996, Mongolia's first president, Ochirbat Punsalmaa, paid an official visit to the UK.
As a part of his visit, he and the Queen watched a ballet concert. In the midst of the concert, someone started to snore. As the sound of snoring became more disturbing, so that everyone, including the queen and the president, tried to find who it was, it turned-out to be one of Ochirbat's bodyguards.
Somewhat worried, the Queen asked Ochirbat if they should wake the poor bodyguard up. In response to that, Ochirbat said, "Oh, please don't. Look at your dancers, pointing to the ballerinas being on demi-pointe (tip-toeing), even they are trying hard not to wake him up."
I did some quick googling, and it seems that president Ochirbat Punsalmaa did indeed meet the queen in 1996. However, I was able to find no reference at all (at least in English language sources) to this alleged ballet concert other than this Quora answer.