Stadler produced a blistering bogey-free eight-under 63 to back up his opening 67, while Karlsson, the leader after the first round was completed earlier in the day, carded a 67 following his opening 63.
The duo share the lead at 12-under-par, while South African Rory Sabbatini is on nine-under through 17 holes of his second round tied with American Chris Stroud after 15.
Sabbatini opened with a 68 and was six-under in the second round when play was suspended due to darkness, while Stroud opened with a 66 and is four-under for his second round.
Venezuelan Jhonattan Vegas (66,68) and Americans Ryan Moore (67,67), Pat Perez (66, three-under through 14) and Jay McLuen (67, four-under through 14) shared fifth at eight-under.
Karlsson, an 11-time winner on the European Tour, is only in the field courtesy of his top 10 finish in last week's McGladrey Classic, where he was a sponsor's invite.
Having gone through a rough patch over the last few years, the former world number six has no status on the PGA Tour and has slipped to 287th in the world. His last win on the European Tour came almost three years ago at the Dubai World Championship.
A week after being in contention but fading to a tie for 10th in Georgia, Stadler is searching for his first career PGA Tour win in his 237th start. The son of former Masters champion Craig had five birdies in his last seven holes.
(Reporting by Ben Everill; Editing by Peter Rutherford)
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