Scotland's Martin Laird also struggled in the windy conditions at Sherwood Country Club in California, bogeying eight holes as he recorded a five-over 77.
Laird was playing alongside Choi and their rounds could not have been more different.
Choi opened with a quintet of birdies and sank another on the 13th to move to six under.
His only dropped shot came on the par-three 15th when he took three to get down from the fringe of the green but he immediately atoned for that with a birdie at the par-five 16th.
Woods also opened brightly, racing to four under par after five holes. A dropped shot on the ninth was regained on the 13th but consecutive bogeys on the 15th and 16th threatened to derail his round.
However, he drained a 15-foot putt on the 17th and two-putted for par on the last to remain firmly in contention.
Woods, without a win in two years, was joined on three under by world number six Stricker, the highest-ranked player in the field, who holed a 25-foot birdie putt on the last.
America's Matt Kuchar was handily placed at four under par on the 16th tee, but he walked off the green with a triple-bogey eight after getting tangled in the rough.
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