When reporting structureThe way in which the parts of a sentence, clause or expression are arranged. E.g.: make somebody + infinitive in "The teacher made me rewrite the composition."s such as wish, would rather, it's (high) time etc., the same rules apply as in the case of reporting second and third conditional sentences.

I’d prefer to stay at home instead. (= I’d rather stay at home.)
I’d prefer to have gone by tram. (= I’d rather have gone by tram. /  I’d be happier if I had gone by tram.)

Note the difference between the sentence above and this one:

subject 1 + would rather/sooner + clause (subject 2 + past perfect)

I’d rather we had never met. (= If only we had never met.)
What would you rather I had written? (= What do you wish I had written?)

subject 1 + would rather/sooner + clause (subject 2 + unreal tense)

would rather/sooner + perfect infinitive

This structureThe way in which the parts of a sentence, clause or expression are arranged. E.g.: make somebody + infinitive in "The teacher made me rewrite the composition." can express preferences in the past.

Would you rather have had pork than veal?
I’d rather not have called the police.

If we delete the word rather, we get the main clauseA clause that could be a sentence in itself. E.g.: I wouldn’t go there if I were you. ("I wouldn't go there") When I arrived he was surfing. ("he was surfing") of a third conditional sentence.

would rather/sooner + infinitive

This structureThe way in which the parts of a sentence, clause or expression are arranged. E.g.: make somebody + infinitive in "The teacher made me rewrite the composition." can express preferences in the present.

I would rather not go to the opera tonight. (= I don’t want to go to the opera.)

Note that *I would not rather go... is not possible in this case.