The ruins in the middle of Pantabangan Dam in Nueva Ecija province are a tourist draw, even as the region swelters in extreme heat
West Indies were 159-3 at lunch on the third day, just 45 runs adrift of England's first innings 204 all out.
Shamarh Brooks (27 not out) and Roston Chase, unbeaten on 13, were at the crease after Brathwaite fall lbw to Stokes for a well-made 65.
West Indies resumed on 57-1, with John Campbell the lone batsman dismissed by England in testing, overcast conditions on Thursday.
But for the first time in the match, Friday's play began beneath sunny blue skies - ideal batting weather.
Brathwaite was 20 not out and Shai Hope three not out, with the pair having scored three hundreds between them in a sensational Test victory at Headingley during the West Indies' 2017 tour of England.
Barbados-born fast bowler Jofra Archer would have had Hope lbw for 16 had a West Indies review not revealed a marginal no-ball.
But Hope was still on 16 when Dom Bess, the lone specialist spinner in England's attack, lured him into a bold drive and Stokes, leading the side in the absence of regular captain Joe Root, held a sharp chance at slip off a fast-travelling edge.
West Indies were now 102-2 after a stand of 59 in a match that marked international cricket's return from lockdown.
It was an encouraging moment for Stokes, who had brought on off-spinner Bess early in Friday's play despite the presence of four seamers.
All-rounder Stokes could have been forgiven for second-guessing himself after deciding to bat first on winning the toss and being involved in the decision to drop veteran paceman Stuart Broad, especially with Brathwaite completing a composed fifty.
But, as he has so often done before, Stokes made a breakthrough when he trapped Brathwaite in front of the stumps.
West Indies captain Jason Holder had led from the front to put his side on top with a Test-best 6-42 on Thursday.
It was the sixth time in his past 10 Tests that Holder, the world's top-ranked all-rounder, had taken five wickets in an innings.
The ruins in the middle of Pantabangan Dam in Nueva Ecija province are a tourist draw, even as the region swelters in extreme heat
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