England were bowled out for 464 in the morning ??? Joe Root left stranded on 182 not out, a shocking run out not helping his fruitless attempt to reach 200 ?????and the tourists then took just two wickets in over two sessions as the Windies closed their second innings on 202-2 in Grenada, a lead of 37.
Brathwaite is still there on 101 not out, with Marlon Samuels alongside him on 22no.
England began the day on 373-6, with Root on 118no. Jos Buttler (13) was the first to fall, stumped when skipping down the pitch to spinner Devendra Bishoo.
Chris Jordan played nicely ??? crunching a couple of fours after benefiting from a slice of luck when an inside edge onto the stumps from his third ball failed to dislodge the bails.
But they were dislodged, by keeper Denesh Ramdin, when Jordan was run out for 16 later ??? although it took third umpire Billy Bowden over four minutes to make the decision.
Stuart Broad???s terrible recent form with the bat continued. He came and went within an over, gloving Bishoo to Devon Smith when attempting the sweep.
Root had brought up his 150 two balls prior to Broad???s dismissal, and with just Anderson???s wicket left, the Yorkshire batsman starting taking a few more risks in search of a second Test double-hundred.
He hit two huge sixes off Bishoo, the first a lovely lofted drive straight down the ground and then an emphatic slog-sweep over midwicket a couple of overs later.
But a brain-freeze from Anderson in the next over brought the innings to an end. Root called for a risky second run looking to keep the strike and despite a wayward throw from the fielder, Jason Holder still had plenty of time to take the bails off with Anderson ambling home and failing to ground his bat. Root looked furious.
England nevertheless had a healthy first innings lead of 165, and Anderson did make partial amends with the ball, Smith (2) playing on when trying to leave an outswinger.
But Brathwaite and Darren Bravo (69) then played superbly and batted through the afternoon session, adding exactly 100 runs in the process as they took tea on 122-1.
Broad finally got the breakthrough, Bravo nicking him behind to Buttler, and that brought Samuels strolling to the wicket. Ben Stokes ??? with whom Samuels has had a running feud throughout the match ??? covered his mouth with his shirt as he walked in.
Samuels and Brathwaite were untroubled, though the latter survived a big appeal and review when Jordan thought he had trapped him lbw ??? an inside edge coming to the rescue.
He reached three figures ??? off his 228th ball ??? just before the close with a cut for two off Anderson, Brathwaite’s fourth century in his last nine Tests.
The pitch is increasingly slow, offering nothing to the quicks or spinners, and ??? barring a dramatic Windies collapse????? a draw looks by far the most likely outcome going into the final day.