Pulis guided Stoke into the Premier League in 2008, where they have stayed ever since, but unappealing football and the lowest points total in the club's short Premier League history saw the 55-year-old axed.
But Coates has thanked Pulis for his work during a seven-year spell at the club, and envisages a difficult task in finding a replacement.
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"I personally regard him as a great friend and will hugely miss working with him.
"We will now begin the difficult task of searching for a successor to Tony. This will be done in private and the club will make no further statement in respect of this search until an appointment is made."
Former Portsmouth and Plymouth manager Pulis is saddened to leave the Britannia Stadium outfit, but accepts the club's decision to replace him.
"Although I am disappointed, I do understand what he means when [Coates] says the board feels a need to take the club in a different direction," read a statement from Pulis via the League Managers Association.
"In leaving I am very proud of what we have all achieved.
"I have enjoyed some wonderful times at Stoke City and nobody was prouder than me when, after 23 years of exile from top-flight football, we gained Barclays Premier League status."