The terms of the Austro-Hungarian dual monarchy were established by negotiations that took place every ten years. It was something of an equal relationship, because both parts couldn't survive without each other. The Austrians and Hungarians might disagree over the terms of the union (and they certainly had many disagreements), but they didn't disagree over its mutual necessity.
As part of the agreement, both sides retained some functions under control of their own crown (royal powers) and some functions were unified (imperial powers). Royal powers were managed by the parliaments, imperial powers were managed by the emperor and required the consent of both parliaments. The army was a royal function, with Austria and Hungary having separate forces. The navy was an imperial function, with Austria and Hungary having to jointly agree to its funding. If the worst happens, the armies would probably fall under control of their respective royal (parliamentary) authority and defend their own territory, while the navy would be more complicated, especially with its large number of Croatian, Slovenian, and Polish personnel.
Basically, you can look at the Confederate war effort during the American Civil War (especially the later part) to see how things might go for Austria-Hungary under severe circumstances.