Red chicken existing without head (9)

I believe the answer is:

cochineal

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'red' is the definition.
(thesaurus)

'chicken existing without head' is the wordplay.
'chicken' becomes 'cochin' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'existing' becomes 'real' (similar in meaning).
'without head' means to remove the first letter.
'real' with its first letter removed is 'eal'.
'cochin'+'eal'='COCHINEAL'

Can you help me to learn more?

(Other definitions for cochineal that I've seen before include "Crimson dye from crushed insects" , "Red food colouring" , "Scarlet food dye" , "Red colour" , "Scarlet dye - in a cloche (anag)" .)

I've seen this clue in The Telegraph.

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