Methods are known for producing an anti-resonant hollow-core fiber which has a hollow core extending along a fiber longitudinal axis and an inner jacket region that surrounds the hollow core, said jacket region comprising multiple anti-resonant elements. The known methods have the steps of: providing a cladding tube that has a cladding tube inner bore and a cladding tube longitudinal axis along which a cladding tube wall extends that is delimited by an interior and an exterior; providing a number of tubular anti-resonant element preforms; arranging the anti-resonant element preforms at target positions of the interior of the cladding tube wall, thereby forming a primary preform which has a hollow core region and an inner jacket region; and elongating the primary preform in order to form the hollow-core fiber or further processing the primary preform in order to form a secondary preform. The aim of the invention is to achieve a high degree of precision and an exact positioning of the anti-resonant elements in a sufficiently stable and reproducible manner on the basis of the aforementioned methods. This is achieved by providing anti-resonant element preforms which have at least one respective ARE outer tube and/or at least one respective ARE inner tube, wherein the ARE outer tube and/or the ARE inner tube is produced using a vertical drawing method without molding tools.