Despite the 1986 song Walk Like An Egyptian by The Bangles, the more widely known name for Walking Like An Egyptian is the Egyptian Sand Dance. This was a staple of British music hall acts in the 19th and early 20th centuries.
Such acts included "My Fancy" (the stage name of Mrs E. Fawn) who was billed in the London Pavilion and the Oxford Music Hall in 1894 as the "Finest Sand Dancer in the World", and the 1930s music hall trio Wilson, Keppel, and Betty, who would perform the Dance entirely deadpan, dressed in fez and long one-piece gown, with sand sprinkled across the stage, to enormous comic effect.
These acts tapped into the flood of popular interest at the time in all things Ancient Egyptian. that was in part due to the discovery of the tomb of Tutankhamun and in part simply because of the popularity of Egyptian archaeology.
The demise of music hall has not stopped people from performing the Egyptian Sand Dance. In the late 1970s, for example, an exhibition of Tutankhamun's funerary cache toured the United States, and once more all things Ancient Egyptian were in vogue, prompting comedian Steve Martin to release a comedy song "King Tut" in 1978, derived from a Saturday Night Live skit, the dance routines for which were the same Egyptian Sand Dance moves. Morecame & Wise reprised Wilson, Keppel, and Betty's routine, with Glenda Jackson as Betty, in one of their comedy shows in 1971.
The Egyptian Sand Dance itself involves attempting to mimic the static postures of the figures to be found on bas reliefs, whilst walking. The most usual stance is to hold the arms crooked, one up and forward and the other down and backward, and to walk by shuffling along, forwards but sometimes also backwards, with one leg in front of the other, attempting to keep all four limbs in a straight line.
Of course, this is purely a music hall invention of the 19th century. As can be seen from Ancient Egyptian statues, people walked the same way in Ancient Egypt they as they do now. The reason for the unnatural postures of the figures on the bas reliefs is to ensure that all four limbs are represented.