Fancy red coat in 1920s style (3,4)
I believe the answer is:
art deco
'1920s style' is the definition.
'fancy red coat' is the wordplay.
'fancy' indicates anagramming the letters.
'red'+'coat'='redcoat'
'redcoat' with letters rearranged gives 'ART DECO'.
'in' is the link.
(Other definitions for art deco that I've seen before include "Do trace (anag) - style of the 1920s and 1930s" , "1920s/1930s style" , "pre-war style" , "Redcoat (anag) -- inter-war design style" , "1920s and 1930s ornamental style" .)