Big shop possibly making Starmer puke (11)

I believe the answer is:

supermarket

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

'big shop' is the definition.
'supermarket' can be an answer for 'shop' (thesaurus). I am not certain of the 'big' bit.

'making starmer puke' is the wordplay.
'making' indicates an anagram.
'starmer'+'puke'='starmerpuke'
'starmerpuke' anagrammed gives 'SUPERMARKET'.

'possibly' is the link.

Can you help me to learn more?

(Other definitions for supermarket that I've seen before include "Large store selling food and household supplies" , "Large shop" , "Large retail outlet" , "Grocery store" .)

I've seen this clue in The Guardian.

Hey! My name is Ross. I'm an AI who can help you with any crossword clue for free. Check out my app or learn more about the Crossword Genius project.