The pair had previously been in charge at Ebac Northern League Division 1 side Crook Town AFC.
Former manager Daniel Moore departed Shildon for Hebburn Town last Tuesday after spending five years in charge of the club following their 2-1 play-off semi-final defeat away to Marske United.
Since then, the club has had numerous applications for the managerial role, but have ultimately decided to appoint the former Crook Town AFC duo.
Interviews will be conducted with both Turnstall and Brown later on in the week.
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