You will never have a subjunctive either in a subordinate clause starting with depuis que, or in the main clause. In your example you must use the passé composé.

Other issues with your sentence:

• break up is rompre and it is conjugated with avoir,
  → ils ont rompu ;

• to express devenir pire we have the verb empirer,
  → a empiré ; which you can use with an amplifier such as beaucoup or considérablment.

  Depuis qu’ils ont rompu son état a beaucoup empiré.

• You are right in saying that this sentence has two different subjects but you are mistaken in thinking it is a reason to use the subjunctive. In French the subjunctive is used to express attitudes or feelings like doubt, uncertainty hope…, but not actual facts, and in your sentence we are dealing with an actual fact.

• Difference between rompre and se rompre:
  An example where rompre and se rompre can be used indifferently:

  Les branches se sont rompues sous le poids des fruits.
  Les branches ont rompu sous le poids des fruits.

  But in your example you use rompre for the people :

  Ils ont rompu.

  And se rompre for the tie that (used to) unite them:

  Leurs liens (leurs rapports) se sont rompus.

• The answer to your initial question: “Is there an imperfect subjunctive tense?”: yes there is, see here but you can’t use it in your sentence and it is generally very rarely used nowadays.