Sweet-chasers consumed snacks (6,6)

I believe the answer is:

cheese straws

I'm a little stuck... Click here to teach me more about this clue!

'snacks' is the definition.
(I have seen 'Savoury snacks ' mean 'cheese straws' so perhaps 'snacks' could also mean 'cheese straws')

'sweet-chasers consumed' is the wordplay.
'sweet-chasers' anagrams to 'CHEESE STRAWS'.
But, I am unsure how this is indicated.

Can you help me to learn more?

(Other definitions for cheese straws that I've seen before include "Twig-shaped nibbles" , "Savoury snacks" .)

I've seen this clue in The Everyman puzzle in The Observer.

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