Fleetwood entered the weekend on five under par - just one off the lead - while McIlroy started Saturday on four under. By the ninth hole, Woods had cut his deficit in half, carding a three-under 33 on the front nine.
Woods has shown glimpses of his old form at times since returning to the tour last December, and he looked in good order as he went out with two birdies, before an erratic inward nine featured three bogeys and just one more birdie.
This has potential to be as fiery a 36-hole finish as the browned-out fairways on the venerable championship course with the reputation as the toughest of all the Open courses.
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But the wind off the North Sea coast continued to cause problems and plenty of big names missed the cut, made at three over par, to ensure they would not take part in the weekend's activities: Dustin Johnson, Justin Thomas and Jon Rahm all failed to make the grade while Justin Rose nailed a birdie putt at the 18th to sneak into Saturday's listings.
Zach Johnson, whose name already is on the claret jug from his playoff victory at St. Andrews three years ago, played in the morning under an umbrella and finished with a 30-foot birdie putt for a 67. The group of 39 includes a Xander (Schauffele) and a Zander (Lombard), the latter a 23-year-old South African with one previous major participation in his life, yet who briefly held a share of the lead in the second round before two bogeys, and said: "Yeah, it was pretty cool". Heading out early, and dealing with cross-winds, the world number eight found himself in as rotten position at the Par 4 second hole.
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It wasn't easy on everyone. Woods asked the gallery to move back because he would have to start it to the right with the golf ball well above his feet.
Rose was one of the early third-round starters, with the overnight leaders aiming to move clear of the Englishman in conditions forecast to still be favourable.
It took a chunk out of Dustin Johnson and Justin Thomas, the Nos. He shot 77 and missed the cut by one shot. Rose is No. 3 and almost joined them until he made an 18-foot birdie putt on the last hole to make the cut on the number.