Both are correctWhen structured with a secondary subject, however they impart minutely different meaningsi.
The former:
I would rather he played with Tim.
implies that the event has happened in the paste. "He" has played[subject] would rather [secondary subject] [predicate] with Tim in the past, and the speaker has an attitude toward that. It is always correct to use past eventtense.
The latter:-Cambridge University Press
I would rather he play with Tim.
-speakspeak
implies that the event is happening in the present. "He" is playing with Tim in the presentTherefore, and the speaker has an attitude toward that present event.of your two examples:
- I would rather he played with Tim.
is the correct form.