Tiger Woods, seeking his first PGA Tour victory since 2009, survived the 36-hole cut by shooting a 2-under 68. He was tied for 31st, seven shots out of the lead.
On an eventful day at the PGA National course, Rose and Tom Gillis emerged as co-leaders midway through the tour's first of four consecutive weeks in Florida.
Rose shot a 66 and Gillis fired a 64, with both players completing 36 holes at 8-under 132. Rose has three tour wins, the last one coming at the 2011 BMW Championship. Gillis has yet to win on the circuit.
McIlroy, ranked second in the world behind Luke Donald, moved into a tie for third at 133 along with Dicky Pride. A victory this week would move McIlroy past Donald and into the No. 1 spot for the first time in his career.
McIlroy faced the same situation last week at the Accenture Match Play Championship, but lost in the championship match to Hunter Mahan.
Harman came to the par-5 18th needing an eagle to shoot 59, but found a greenside bunker with his second shot. He almost holed his third shot, but it ran 5 feet past the hole and he missed the putt coming back. He had eight birdies and an eagle in his round.
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