I kind of wonder if this is evolving these days, though. While support for the concept of a civic people's army is high, the combat units are becoming a bit more ideological I guess in ways that they were not formerly (the issue of females singing, for example, has become more prominent), and this is not just a rank and file issue anymore but also with the junior officer corps. Service evasion may be a taboo but that doesn't mean it isn't happening more and more, just in less ostentatious ways. The volunteer army is probably inevitable.My point exactly. The IDF is an effective cross-section of the non-Arab, non-Haredi population. You won't get the kind of separation between the military and civilian population needed for the military to back a coup. If the government is that unpopular, it would collapse. The only scenario I can come up with is some sort of insane election rigging scenario or the government trying to disenfranchise large swaths of the electorate, both of which are borderline ASB.
As for a coup, if it was to happen, it would probably come from internal high level type stuff that would represent a coup but perhaps not actually be one. The relative popularity gap between Moshe Dayan and Levi Eshkol in 1967, for example, could lead to a sort of de facto sidelining of Eshkol as PM during the crisis, and could if things get nasty enough be considered a coup of sorts.