They're both at 7-under par after being part of a five-way tie for the lead following round one.
Appleby made a 16-foot putt to save par on the 18th and only has two bogeys in the tournament. Choi's 11-foot birdie putt on the 16th hole took a full lap around the rim of the cup before falling in.
Appleby and Choi enjoy a 3-shot lead over a group of three led by Billy Andrade.
Tournament host Tiger Woods played himself back into contention with a 4-under 66 that tied Mike Weir for the best round of the day. He's six off the lead, while Weir is five shots back.
Phil Mickelson shot a 73 and missed the cut.