Both players advanced to ten under par after ten holes at Jumeirah Golf Estates, with Race to Dubai leader Martin Kaymer, Spain's Alvaro Quiros and Thailand's Thongchai Jaidee a shot further back.
The English Ryder Cup teammates had been three shots apart when Fisher bogeyed the second and Poulter birdied two of his first three holes.
Poulter, who went 56 holes without dropping a shot last week, was furious with himself when his run of bogey-free holes ended at 40.
He three-putted the fifth and when Fisher birdied the long seventh the gap between them was back to one.
And when Poulter bogeyed the tenth it was all square at the top.
Meanwhile Graeme McDowell conceded defeat to Martin Kaymer in their battle to become Europe's Number One.
But the Ulsterman also said after a third round 69 that a season of incredible highs - the US Open Championship and The Ryder Cup, of course - has left him convinced he can go on to become World Number One.
Needing a third place finish at the final event of the year-long Race to Dubai, McDowell knew he had to produce something really special when he resumed in 42nd place.
His three under par score lifted him into 29th spot, but with Kaymer in contention at the tournament the Ulsterman commented: "Barring the impossible Europe's Number One is not a reality for me any more.
"Martin is definitely going to do it unless I shoot 55, which unless one of you boys (a handful of reporters) putts for me isn't going to happen.
"But what I've learnt this season is that I'm good enough. I know what room I have for improvement, but it was great to see that my best is good enough to compete.
"Two or three years ago the number one seemed untouchable, but I believe if I work hard enough I can become the best player in the world."
McDowell felt his chances ended when he three-putted the short fourth and then missed birdie chances on the next two greens.
"I was just waiting for the taxi to turn up and take me home at that point," he said.
"The course has not shown me much love. It's not my cup of tea - it frustrated me last year and is continuing to frustrate me.
"Sometimes you've just got to wave the white flag. I tried my heart out and it just didn't happen, but I'll have a cold beer and look back on a great season."