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According to Esquire, " the team on First Man played it strictly by the book, and, aided by the boffins at Omega's headquarters in Bienne, Switzerland, set out to equip the cast in period-correct Speedmasters Production began in the summer of 2017, and throughout the filming schedule it used Omega archive watches and specially made replicas, all consistent with the timeframe of the movie More than 10 period-specific Speedys, in two different references, were made for Gosling and the Apollo 11 team.
The reason for the two references is that the ST 105.003 and the ST 105.012 were both worn by Armstrong first in training and then for the actual Moon mission. . Source