London gallery with a little less taste (4)

I believe the answer is:

tate

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'london gallery' is the definition.
(I've seen this in another clue)

I don't understand how the rest of the clue works.

Can you help me to learn more?

(Other definitions for tate that I've seen before include "One of four art galleries in England" , "Lord Sugar" , "Modern and Britain galleries in London" , "Gallery (or two!)" , "famous art gallery beside London's Thames" .)

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

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