The Straw Hat industry was a very important home industry in the St Albans and Luton area (which is why the Luton Football team is called the Hatters).
Straw was split along its length and plaited into long strips by women and children at home and sold on to dealers, with others using the plait to make straw hats. Very large numbers of people were involved and the 1881 census for St Albans Abbey Parish includes some 500 people who are clearly linked to the industry. Occupations mentioned include Bleacher of Straw Plait, Blocker, Bonnet Sewer, Brazilian Hat Manufacturer, Clerk In Straw Factory, Commercial Traveller Straw Hats, Felt And Straw Hat Manufacturer, Hat & Bonnet Sewer, Hat Blocker, Hat Sewer, Hat Stiffener, Hat Trimmer, Plait Basket Maker, Plait Maker, Plait Miller, Plait Sewer, Straw Basket Maker, Straw Finisher, Straw Hat Blocker, Straw Hat Finisher, Straw Hat Machinist, Straw Hat Maker, Straw Hat Manufacturer, Straw Hat Packer, Straw Hat Sewer, Straw Hat Stiffener, Straw Hat Trimmer, Straw Plaiter, Straw Sewer, Straw Tailor.
Initially it was impossible to dye straw but I believe one of the first successful dyes was a red dye made from Brazil wood. While I have not seen a formal definition I am pretty sure that a Brazilian Hat Manufacturer made straw hats incorporating coloured straw.
From "Hertfordshire - Little Guide -1903":
The wages earned by peasant girls and women in this employment were formerly high; 100 years ago a woman, if dexterous, might earn as much as £1 per week, but the increase in machinery and the competition from foreign plait has almost destroyed this cottage industry in some districts. During the last twenty years several large straw hat manufactories have been erected in St. Albans, and the trade enlarged, although the conditions of production are altered.