εἰς ὑπακοὴν πίστεως [Stephanus TR 1550, undisputed]
... having been made known to all the nations for obedience of faith -- YLT
'Obedience' in this passage is υπακοήν 'hearing under' or 'subject to hearing' or 'hearing in subjection'. However one expresses that in English, there is an agreement between 'underness' and 'hearing'.
'Obedience' in this passage is dative singular and 'faith' is genitive singular (Bagster's Analytical Greek Lexicon p326 and p413) and the preposition is eis meaning to or towards or unto.
The translation can only be 'to (or unto) obedience of faith' and not otherwise. The obedience is that which is originated by or produced by or caused by - faith. That is the effect of using the genitive case.
This is the hearing of faith, not the exerting of flesh in the pursuit of a self-induced righteousness by way of legal works.
It is a very similar word to that used in Philippians 2:8 ὑπήκοος - became 'obedient' unto death, even the death of the cross.
The foundation of the faith, wrought in Christ, was a matter of faithful obedience, a hearing aright and a responding thereto.
Thus also the reception of that redemption is by the hearing of faith also.