Local runner joining army unit is no monarchist (10)
I believe the answer is:
republican
'no monarchist' is the definition.
'local runner joining army unit' is the wordplay.
'local runner' becomes 'publican' (person who runs a pub or 'local').
'joining' is a charade indicator (letters next to each other).
'army unit' becomes 're' (Royal Engineers).
'publican' after 're' is 'REPUBLICAN'.
'is' acts as a link.
(Other definitions for republican that I've seen before include "in foreign party" , "American in party" , "American politician" , "anti-monarchist" , "Member of US political party" .)