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Barnes, 27, was sent off as the Dabbers were beaten 4-1 at Cantilever Park in last season's play-off semi-final between the two clubs.
He becomes Paul Carden's third new signing of the summer, following Alex Byrne and Scott Sephton in moving to the play-off winners.
The Liverpudlian came through the ranks at Stockport County, and has also played for Southport, Northwich Victoria and Marine.
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