Thank you to Bob Rutemoeller and Mary Sue Ittner (from the Pacific Bulb Society) for identifying my mystery plant.
The mystery plant has been identified as Phormium tenax or New Zealand Flax. According to Wikipedia
Phormium tenax (harakeke in Māori, New Zealand flax) is an evergreen perennial plant native to New Zealand and Norfolk Island that is an important fibre plant and a popular ornamental plant. The plant grows as a clump of long, straplike leaves, up to two meters long, from which arises a much taller flowering shoot, with dramatic yellow or red flowers.
The fibre has been widely used since the arrival of Māori to New Zealand, originally in Māori traditional textiles and also in rope and sail making after the arrival of Europeans up until at least WWII . . . Read more . . .
There are also different cultivars which have different coloured leaves including one with bronze coloured leaves (probably the one you have in your garden, Reggie). See the Wikipedia link for more information.
Thank you to everyone who helped to try and identify the plant!