Cryptic crossword clue
Dirty setter’s embracing posh Doctor of Divinity 5

The most common and most likely answer to this crossword clue is the 5 letter word MUDDY

I've seen this crossword clue in The Guardian
M
U
D
D
Y

Buddy explains!

The straight part
We know that "Dirty" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dirty = muddy.
The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.
This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.

What does MUDDY mean?

MUDDY verb
  1. To get mud on (something).
  2. To make a mess of, or create confusion with regard to; to muddle.
MUDDY adjective
  1. Covered with or full of mud or wet soil.
  2. With mud or other sediment brought into suspension, turbid.
  3. Not clear; mixed up or blurry.
  4. Confused; stupid; incoherent; vague.
  5. Soiled with feces.

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