English and Karlsson were 15 under overall with 11 holes left in the round.
English matched the lowest round of his PGA Tour career in the morning with a 9-under 62 on Mayakoba Resort's El Camaleon course. The 24-year-old former Georgia star won the FedEx St. Jude Classic in June for his first PGA Tour title.
"The last three weeks, last two tournaments, my ball-striking has been unbelievable," Harris said after the second round. "Finally, this tournament, my putting has come around. I changed putters and went back to my old Ping putter that I played with in college. Just brought back some good memories."
Karlsson played 36 holes Friday, shooting 63-67. He birdied Nos. 3-5 in the third round.
"It was nice to knock in a few birdies," Karlsson said.
The 44-year-old Swede, an 11-time winner on the European Tour, tied for 10th last weekend in the McGladrey Classic -- playing on a sponsor exemption -- to earn a spot in the field this week.
"It's all good to be here," Karlsson said.
None of the 78 players who made the cut were able to finish the round.
More than 4½ inches of rain hit the course from Wednesday afternoon to Friday afternoon. Because of the wet conditions, players are using preferred lies.
"It's unbelievable, I think, that we're playing golf at all," Karlsson said. "The amount of rain we've had, the golf course should be unplayable. So, to just play is fantastic. It's drying out nicely day by day."
"The course is in spectacular shape considering how much it's rained," Quiros said. "It's tough to think that we can play a four-round tournament with all the rain that we've had. It's playing longer. I had the opportunity to play here in 2008, and it was different conditions back then. I can say that it's just playing longer."