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Earlier, Alzarri Joseph (3-50) and Jayden Seales (3-50) led the West Indies bowling effort, and debutant Anderson Phillip (2-30) and Kyle Mayers (2-35) backed up brilliantly.
It was yet again a struggle for Bangladesh as only Litton Das with 53 and Tamim Iqbal with 46 made a score of note.
West Indies lead the two-match series 1-0 after a seven-wicket win in the first Test in Antigua last week.
