City were let down by slipshod finishing as they slipped to defeat at Bognor in Ryman League Division 1.
Keeper Alan Foster's only real action was to pick the ball out of his net twice in the second half, by which time City should have had the points wrapped up.
Home keeper Coleman denied Dwaine Strong early on and both Danny Wise and Shaun Wimble went close to putting the visitors ahead.
After Chris Organ had cleared a rare home threat with Foster beaten, Wise blazed over in the 25th minute after a one-two with Mark Druce.
Coleman tipped over a good effort from Wimble and as City continued to squander chances, Strong shot over after a lovely reverse pass from Druce, and on the stroke of half-time, Wise finished weakly when he shot wide with only Coleman to beat.
City's misses were to prove costly. In the 52nd minute, Birmingham drove low past Foster from 12 yards, and they were two down in the 67th minute. Wimble gave the ball away at a City corner and a long clearance allowed Mulhern to run from halfway to round Foster.
Druce twice set up chances which the home defence survived and with five minutes left, Wise shot straight at Coleman after a good cross by Strong.
There was still time for Coleman to make the save of the game to keep out a cracking 25-yard drive from Wimble.
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