Reportedly, wood supporter behind very rotten tree, back part (6,8)

I believe the answer is:

lumbar vertebra

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'back part' is the definition.
(I know that lumbar vertebra is part of back)

'reportedly wood supporter behind' is the wordplay.
'reportedly' shows a homophone (sound like).
'wood' becomes 'lumber' (I've seen this before).
'supporter behind' becomes 'vertebra' (I can't explain this - if you can you should believe this answer much more).
'lumber' sounds like 'lumbar'.
'lumbar'+'vertebra'='LUMBAR VERTEBRA'

'very rotten tree' is the link.
This may not be correct. Some or all of it may be part of another bit of the clue.

Can you help me to learn more?

I've seen this clue in The Guardian.

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