Pia's pet sure tears up delicate stuff (6,5)

I believe the answer is:

tissue paper

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'up delicate stuff' is the definition.
'tissue paper' can be an answer for 'stuff' (tissue paper is a kind of stuff). I am unsure of the 'up delicate' bit.

'pia's pet sure tears' is the wordplay.
'tears' indicates anagramming the letters.
'pias'+'pet'+'sure'='piaspetsure'
'piaspetsure' with letters rearranged gives 'TISSUE-PAPER'.

Can you help me to learn more?

(Other definitions for tissue paper that I've seen before include "Kleenex" , "There may be a valuable article in it!" , "Wrapping for delicate objects" , "delicate sheets" .)

I've seen this clue in The Irish Independent.

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