Ministry put female in igloo? (4,6)

I believe the answer is:

home office

This clue might be a double definition.

'ministry' is the first definition.
(I know that Home Office is a type of ministry)

'female in igloo?' is the second definition.
The definition and answer can be both man-made objects as well as being singular nouns.
Perhaps they are linked in a way I don't understand?

'put' is the link.

(Another definition for home office that I've seen is " Government department".)

I've seen this clue in The Independent.

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