Predate horribly tedious bureaucracy (3,4)
I believe the answer is:
red tape
'tedious bureaucracy' is the definition.
'predate horribly' is the wordplay.
'horribly' indicates an anagram.
'predate' anagrammed gives 'RED-TAPE'.
(Other definitions for red tape that I've seen before include "Bureaucratic delay" , "unnecessary paperwork" , "(Obstructive) bureaucracy" , "European regulations?" , "Rigid formality - bureaucracy" .)