The Tigers were 157 for four against hosts Ireland when the players were forced off the field, with Tamim 64 not out from 88 balls.
The opening batsman was unable to play the full range of his attacking strokes because his side found themselves in trouble at 70 for four. But in partnership with Mahmudullah he led an impressive recovery, with the latter 43 not out when the rain came.
Peter Chase, who had been Ireland's most consistent bowler in last weekend's one-day internationals in England, was again their best bowler, striking twice in four balls to dismiss Soumya Sarker, caught behind from the fifth ball of his first over, and then Sabbir Rahman, to only his third ball, after a rash pull shot was well caught at long leg by the diving Tim Murtagh.
Shakib Al Hasan, captain for the day while Mashrafe Mortaza serves a one-match ban, also scored quickly but he was Chase's third victim when the bowler returned for a second spell, to a second catch by wicket-keeper Niall O'Brien.
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But Tamim and Mahmudullah finally hauled the world's seventh ranked team back into the match and both were looking dangerous after adding 87 from 99 balls.
There was a blow for Ireland on the morning of the match, when leading batsman Ed Joyce was ruled out with a stiff back. Stuart Thompson was brought in as the extra seamer on a green Irish wicket.
Joyce hopes to be fit for Ireland's next game against New Zealand on Sunday while Bangladesh will return to action next Wednesday for their game against the Kiwis, when Mortaza will be available.