Tense outfit, American, obscure - one’s involved in multiple partnerships (9)
I believe the answer is:
trigamist
'obscure one's involved in multiple partnerships' is the definition.
I can't judge whether this defines the answer.
'tense outfit american' is the wordplay.
'tense' becomes 't' (grammatical abbreviation).
'outfit' becomes 'rig' (synonyms).
'american' becomes 'amist' (I can't explain this - if you can you should believe this answer much more).
't'+'rig'+'amist'='TRIGAMIST'
Can you help me to learn more?
(Another definition for trigamist that I've seen is " One with three spouses".)