Bangladesh still trails by 344 runs despite Tamim's effort, which included eight boundaries and one six. Mushfiqur Rahim was 33 not out.
West Indies declared its second innings on 383-5 earlier Tuesday when Darren Bravo was dismissed for 195, trying to raise his maiden test double century with another six.
Shivnarine Chanderpaul finished unbeaten on 59. He reached his half century with consecutive pulled boundaries behind square leg from Shahadat Hossain in an over that cost Bangladesh 16 runs.
Before tea, Bangladesh lost two quick wickets — Imrul Kayes caught by Kirk Edwards off Fidel Edwards for nine and Shahriar Nafees caught and bowled by skipper Darren Sammy for 18.
In the final session, Raqibul Hasan (17) was caught by Sammy off a Marlon Samuels delivery.
Earlier, Bravo resumed day four on 100 after reaching his maiden test century on the last ball of the previous day. He batted early on with nightwatchman Kemar Roach, who added only eight runs to his overnight total before he was dismissed by Sohrawardi Shuvo (3-73) for 12.
Bravo added 143 for the fifth wicket with Chanderpaul and had just clobbered Shuvo over long-on for six — his fifth of the innings — when he top-edged another attempted slog and was caught out five runs short of his 200, bringing an end to his 297-ball innings.
It was an impressive 10th test for Bravo, a 22-year-old left-handed batsman who posted six half centuries in his first eight tests.
On Monday, Kirk Edwards made 86 to get the West Indies' second innings off to a decent start after Bangladesh was dismissed for 231, conceding a 124-run first-innings deficit.
