No quisiera que se rompa el record
 

Please explain the grammar here, which contains a past subjunctive - quisiera - with a present subjuntive - rompa:

"No quisiera que se rompa el record"

Personal opinions and feelings are accompanied by the subjunctive.
In this case, "no quisiera que" is used to say what I wouldn't want to happen, and "rompa" shows the possibility that things will happen despite my wish.

I'll never agree with those explanations (which doesn't exist in Spanish).

The sentence is "no quiero/quisiera" [conveying different nuances] "que se rompa el récord" simply because ":bad:no quiero que se rompe el récord" makes no sense in Spanish.

The reason is English indicative and Spanish indicative are quite different. When you use indicative in Spanish, the action expressed in the verb is being carried out. In Spanish you say "llueve" because it's indeed raining. You can't say ":bad:puede que llueve" because it's not raining -it's almost that simple- so we have subjunctive to refer to actions that for some reason are not straightforwardly happening "puede que llueva" as with Spanish subjunctive it may rain or not.

If you tried to say ":bad:no quiero que se rompe el récord" there would be two actions that are happening in that sentence: you want something and the record is broken, altogether. A native speaker would try to make sense of it by hearing "no quiero, que se rompe el récord" (I don't want to, because the record ends up broken). If you don't want the record to broke, you have to do something exactly like adding "to" in front of the verb in English. In Spanish, subjuctive is handy to that and many other tasks, so by "no quiero que se rompa el récord" we understand that "you want", and the record is not breaking but you want it unbroken.

Alec: Grammatically, the personal opinions that involve how the speaker feels about something, like in this case "no quisiera", need the subjunctive. This is a different case from "puede que llueva", which expresses uncertainty.

Similar to the OP's case are the expressions:
- Alegrarse de que
- Lamentar que
- Preocuparse de que
- Dolerle a alguien que
- Tener la esperanza de que
...

I understand, but my question is primarily why the past subjunctive is mixed together with the present subjunctive in this example.

It might sound better with "que se rompiera" so both verbs agree. However, on one hand, "no quisiera" nuances "no quiero", which feels like a whim; and "rompiera" sounds far from the possibility that the record is broken (which the speaker seems to find close).

excellent, thanks

Here you need to say "to break" (not to broke).