The EVO-STIK League Southern Division One South & West encounter was suspended on the half hour mark with the score at 0-0 as fog filled the Winterfield Road stadium.
Officials had hoped the fog might lift after a 15-minute break but the conditions worsened and the fixture was abandoned soon afterwards due to poor visibility.
Fan Liam Brown, who runs the Salisbury Supporters' Club Twitter account, was at the match and told salisburyjournal.co.uk: "From the start, it seemed to be in doubt, but leading up to kick off it did start to clear.
"It was looking like it was slowly getting better, but suddenly the fog truly descended, and that's when the referee called the players off. As we left, it does seem like the right decision, as visibility was just getting worse and worse.
"It is annoying that we'll have to make the trip again, as it means another matchday added to the busy fixture list. But it's very good of Paulton Rovers to have announced that the replay will be free admission. A very good gesture on their part."
Subject to league approval, the clubs are hoping to meet again on Tuesday 10 January.
Follow the EVO-STIK League Southern on Twitter at @SouthernLeague1
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