Mann, who missed out against Argentina, returns to the blind-side flanker role alongside Jac Morgan and Aaron Wainwright.
Reffell is set to make his first international appearance for a year, and his first selection by Tandy, after being left out of the previous two campaigns.
“We feel we’ve got depth in the back row,” said Tandy.
“Manny (Alex Mann) was excellent against Fiji, but he hadn’t played a lot of minutes so now we’ve got a good opportunity to freshen up the pack.”
Cardiff lock Williams, who will partner Adam Beard, makes his first Test start for a year after overcoming a serious leg injury suffered at the beginning of the season,
“It’s brilliant for Teddy, how hard he’s worked after a horrific injury and the way he has come back,” added Tandy.
“He had a real impact when he came off the bench last weekend, so it’s pleasing and well-deserved.”
With Carter sidelined, Wales face the Springboks without the lock pairing that finished the Six Nations with Exeter captain Dafydd Jenkins missing the summer tour because of a shoulder operation.
Carter forced his way into the starting line-up for the final three games of the Six Nations and continued in the side for the uncapped fixture against the Barbarians and Nations Championship games against Fiji and Argentina.
The absence of the 21-times capped Dragons forward further stretches the second-row resources.
Flanker Plumtree covered the second row in the fixture against Fiji and Thomas was called to South Africa and now finds himself thrust into the 23-man squad.
It’s a baptism of fire for Freddie,” said Tandy.
“He was unlucky not to be in the original squad. He’s come in, trained well and it’s a great opportunity for him on the weekend.”
Wales complete the season against a team that defeated Tandy’s side 73-0 at the Principality Stadium in Cardiff in November 2025.
Wales have won three and lost eight of the 11 internationals so far in Tandy’s first season in charge, with a victory and defeat so far in the Nations Championship.
“South Africa are world champions and have loads of depth,” said Tandy.
“For us, it’s about putting our game out there. We want to see a real physical edge.
“We’re going up against probably the most physical team in the world.
“It’s a great opportunity to test ourselves against the best team in the world.”