Fisher had seven birdies and a bogey while Gallacher scattered six birdies around his scorecard.
They are one stroke up on George Murray, who had four consecutive birdies in one stretch en route to a 5-under 66. A late bogey kept him from joining Fisher and Gallacher in first.
Five golfers shot 4-under 67 and seven others are three strokes off the lead.
Fisher, playing the second nine at the Oceanico Victoria Golf Course first Thursday, opened with two pars and then posted three birdies over Nos. 12-15. A bogey at the 208-yard, par-3 16th slowed him before he opened his second nine with birdies at 1, 2, and 3. A birdie at No. 5 moved him into the lead.
Gallacher, also starting the back nine, had birdies at 11, 15 and 17 to make the turn at 3-under. Birdies at 2, 5 and the 188-yard, par-3 eighth pulled him into the tie with Fisher.
Jessica Simpson shares three-way kiss with friends in photo
Interpol investigating stolen passports on missing flight