Having been exposed, Greek royal is fuming (7)

I believe the answer is:

reeking

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'fuming' is the definition.
('reek' can be a synonym of 'fume')

I cannot understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for reeking that I've seen before include "Having a foul smell" , "With a very unpleasant smell" , "Stinking" , "Emitting smoke" , "Foul smelling" .)

I've seen this clue in the Sydney Morning Herald.

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