PAU, France — On the eve of the Chanel cruise 2027 show in Biarritz, the looms at the ACT3 workshop in the nearby city of Pau were churning out not just the brand’s signature tweed, but a brand new fabric developed especially for the resort collection. Creative director Matthieu Blazy updated the traditional striped linen of the Basque region of southwest France by adding flecked horizontal bands, using the fabric to give the signature Chanel skirt suit a sporty twist. A worker stood by to supervise the machine, making sure that no knots ruined the smooth surface of the densely woven cotton. Stringent quality control is non-negotiable at ACT3 — an acronym that stands for Activité Création Tissage, or Creative Weaving Activity in English. The company has worked hand-in-hand with Lesage, the embroidery and weaving specialist owned by Chanel , for close to 25 years, and officially became part of the company in 2014.…