The move to CEO and president leaves his chief operating officer role vacant, while last week's exit of Darren Blackhurst means the merchandising team is without a trading director.
Blackhurst's departure would be a major loss for Asda, analysts predicted.
"The people on Asda's board know the strategy and have been behind it for some time," said Verdict analyst Malcolm Pinkerton. "But to lose someone with that level of experience in buying who is that well regarded is quite a blow. If they want to be top in general merchandise, they might look to the likes of Matalan or Primark."
Jonathan Flynn, head of retail practise at Interim Partners, added: "The people who were rumoured to take over as CEO were all high-profile individuals within the business and I can't think of anybody in the lower ranks ready to step into the COO role."
Asda said it would announce Blackhurst's successor within "a matter of weeks". No decision has yet been made on a successor to Clarke.
Nick Bubb of Arden Partners said Walmart had "played it safe" by giving Clarke the reins.
"Asda has a strong culture and it can be disruptive to bring on someone from outside the team. He's a capable man but not a Walmart stalwart. I thought they'd parachute in one of their own. It begs the question of what happens if Clarke can't turn it around. It could be a shortlived appointment. It's a tough shop to turn around, given the strength of the competition."