Obsessive union man mostly a pig when sloshed (10)
I believe the answer is:
polygamist
'obsessive union man' is the definition.
(union can mean a marriage)
'mostly a pig when sloshed' is the wordplay.
'when sloshed' indicates anagramming the letters.
'mostly'+'a'+'pig'='mostlyapig'
'mostlyapig' with letters rearranged gives 'POLYGAMIST'.
(Other definitions for polygamist that I've seen before include "Person multiply married" , "Great family man" , "One multiply-spliced" , "One with multiple spouses" , "not being single-minded" .)