The British set up many gentlemen's clubs for army officers and civil servant expats in the colonial days, including one in Burma's capital of Rangoon. (They ruled over the country, now called Myanmar and its major port Yangon, from 1824 to 1948.)
Their meeting place was called The Pegu Club, named after the Pegu (now Bago) River which flows from the Pegu mountain ranges down south, ending in Yangon.
Rudyard Kipling stayed a night on his way to Mandalay.
The bar at each of these British clubs is said to have had a unique signature cocktail, and the cocktail at The Pegu Club was named after the establishment.
The club is said to have been built in the late 1800s, but there is no clear evidence of when the cocktail was invented, or if club members even called it The Pegu Club. The first record of its existence is in a 1927 book named Barflies and Cocktails by Harry McElhone, the Scottish bartender of Harry's New York Bar in Paris. The bar was frequented by international travellers, and perhaps McElhone had heard about the cocktail through one of his visitors.
Other than the name, and perhaps the citrusy, fresh, warm-weather-friendly flavours, the beverage has little to do with Myanmar. Its main ingredient is gin, about as quintessentially British as one can get. The other ingredients are lime juice, orange curacao, Angostura bitters and orange bitters.
In the aftermath of the Japanese invasion of Myanmar in the second world war, the British left the country and the club. The building which housed the club has since been abandoned and with it, it seems, any record of the cocktail.