This answer draws from the answer by michael_timofeev, but adds some clarification.
Here we have two separate clauses separated by a semicolon. As with most sentences of this type, each clause can function as a sentence on its own as is not affected by the tense of the other sentence.
So, focussing on the second clause/sentence, we have either:
- "It (being out of the public eye) would mean the death of his political career."
- "It (being out of the public eye) would have meant the death of his political career."
The first sentence is in the "past subjunctive". This is a misleading name as this almost never refers to past events, rather to present or future events.
Thus, the implication of sentence 1 is that the choice of whether to be in or out of the public eye stands before "him", and that one of these choices will ruin his career if he takes it. As stated by michael_timofeev, this would make more sense if the first clause/sentence was in the present tense (with "can" instead of "could").
The second sentence is in the past perfect subjunctive. This is referring to a timepoint in the past, at which "he" had to make the decision on whether to remain in the public eye. After this reference timepoint, his political career might have been ruined. Because this refers to the past, it pairs well with the first clause/sentence (also in the past tense, with "could"), so would be preferable here.