The company certainly had a long history on this site. Elizabeth Tull, 'Net, Twine & Line-maker' appears at the address in Kent's Directory of 1794; the current building dates from 1880. The actual manufacture was carried out in Globe Lane (now Globe Road), Mile End. (An Old Bailey case of 1818 concerned the theft of 18lb of hemp from the 'manufactory'. The thief was transported for seven years.)
Tull seems to have enjoyed a reputation for quality. In 1827, one WP Richards wrote to the Literary Gazette with an account of his trial of an anti-rot treatment for rope. The experiment used 'new cord, of the very best quality, sold by Mr. Tull, in Fenchurch Street.' A few years later, Tull were among the exhibitors at the Great Exhibition of 1851, showing 'twine, fishing lines, cords, ropes, casting nets, & c., made of different materials'.
Today, the company seems to have long disappeared. However, its representatives carry on fishing the first floors of Fenchurch Street, almost unnoticed by passers-by.