Moeen Ali dismissed Kraigg Brathwaite for 95, ending the West Indian opener's resistance at Headingley.
West Indies went to tea on 199-3, still 123 runs short of the victory target of 322. That total may be just out of reach, but the West Indians do have a good chance of drawing the game and keeping the three-match series alive.
With seven wickets left, the West Indians need to survive 35 overs in the final session, the last 16 of them likely against the new ball. The gamble for the tourists is whether to risk losing wickets in going for the run chase.
Five short of three figures, Brathwaite went for a flashing cover drive off Ali in the second-last over of the session and edged to Ben Stokes at slip. Had he reached 100, the 24-year-old Brathwaite wouldn't have just been the first man to make 100 in both innings of a test at Headingley, but the first in over 500 first-class games held at the Leeds ground.
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It didn't quite happen. England dropped Brathwaite twice earlier in his innings but Stokes clung on to the third chance to ease England's frustration.
If West Indies reaches the target, it will be the second-highest fourth-innings score ever to win a test at Leeds.
In a match that has swung back and forth, England was on top in the first session of Day 5 when opener Kieran Powell was caught by Stokes at slip off Stuart Broad for 23.
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