Mickelson topped the leaderboard for most of the day and Watney took his turn alone in first place during the final stretch before both players fell back with double-bogeys to head into Sunday's final round knotted at eight-under-par 208.
Watney shot a one-under 71 and Mickelson a 73.
Fellow-American George McNeil bogeyed the 18th to fall out of a three-way tie for the lead and was alone in third place on seven-under 209.
The stumbles by the leaders brought a slew of other contenders into the frame.
Two strokes off the pace at six-under-par 210 were Britons Lee Westwood (72) and David Lynn (71), Australian John Senden (67), Robert Karlsson of Sweden (69) and Americans Ryan Moore (68) and Derek Ernst (72).
Five players were another shot away, including world number two Rory McIlroy of Northern Ireland, who struggled with his putter on the way to posting a one-over 73 for 211.
A gamble that backfired, a shanked tee shot and an approach that bounced off the head of a spectator highlighted a comedy of errors over the last holes.
Mickelson, the overnight leader by two strokes, led by one shot heading to the par-five 15th.
After a pulling a drive just outside the cart path and behind a tree on the right side of the fairway, the big left-hander decided to steer a metal-wood shot around the tree but it took off straight out of bounds.
"The second shot should not have been a problem," explained Mickelson after the double bogey. "If I had pulled (out) the driver like I did the second time, then it would have cut around, no problem.
"But I tried to do it with a three-wood and it shot straight and went out of bounds. I thought the driver might go in the bunker and I wanted to play it short."
A bogey at the par-four 16th cost Mickelson another stroke.
Mickelson was set up perfectly in the fairway but hit his second shot left of the green and off the head of a spectator down onto the 17th tee. He tried to run the ball through thick grass over a hill back to the green but got caught in the rough.
The bogey left him tied for the lead as Watney squandered his good fortune by shanking his tee shot at the par-three 17th on his way to a double-bogey five.
"I'm still in shock about 17. That's just a terrible feeling. I hope it doesn't happen again," a sheepish Watney said. "Ever.
"But the big picture, I'm tied for the lead."
Players will be grouped in threesomes and start earlier than usual on Sunday as forecasts call for possible heavy rains overnight that could last until Monday.
"It means we're gonna have to be even more patient," said Watney. "It's going to be a long day. But it will be fun. This is what you play for. We kind of let a lot of guys back in. It will be a shootout and I look forward to it."