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However, Dan Lafferty drew the Bards level shortly after the visitors went down to 10-men at the Victoria Ground.
Sporting reclaimed the lead through Jake Jervis’s penalty only for the Bards to score a second equaliser moments later through Kory Roberts.
The result means Stratford sit 17th in the Southern League Premier Central table, three points clear of the relegation zone.
Stratford went close with the game’s first attack as Adam Jasper hit the post after good work from Fin Brennan down the left.
But Sporting opened the scoring after just four minutes when Reeves helped a corner into the net from the first set-piece of the match.
Bards goalkeeper Jake Weaver kept the visitors in the game as he made a string of saves the best of which saw him keep out Jamille Matt’s instinctive effort.
And the Rouslers went close again moments before the break as Kieron Freeman saw two shots cleared off the line in quick succession.
The game looked to have turned in Sporting’s favour after the break on 55 minutes when Stratford’s Critchlow-Woyo, on loan from Coventry City, received a straight red card for a challenge on Simeon Maye.
However, the Bards defied the odds to draw level three minutes later through Lafferty’s composed finish from the edge of the penalty area.
Sporting went back in front on 72 minutes after Weaver fouled Billy Shaw in the box and Jervis fired the resulting penalty into the bottom corner.
But the lead lasted just two minutes as Roberts volleyed in Reece Flanagan’s corner from close range to stun the home side a second time.
Stratford are next in action against Hitchin Town on Saturday, October 31 with kick-off at 3pm at the Arden Garages Stadium.
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