One important component of Austria-Hungary was that it was a dualist system. The Magyars were left to their own business while the other half of the monarchy was left to the Germans with some primitive autonomy for the other regions, such as Galicia. Yet as early of 1871 there were attempts to even out the dualism by attempts of federalization in the Austrian half of the monarchy. One of those attempts were the
Fundamental Articles of 1871. They would've replaced the Austrian House of Lords with a Senate, and given the Kingdom of Bohemia relative autonomy within the Austrian portion of the empire with her own diet and the acceptance of Czech as an official language alongside German. Essentially it would be amounting to Trialism, as it'd give Bohemia similar rights as Hungary. There was opposition not only from the Magyars and the Hungarian Prime Minister, but the German Liberals, who opposed any equalization in Bohemia.
What if the Articles had been accepted by Franz Josef? Was it an arrangement set to fail, or would him accepting the articles despite backlash give the monarchy some breathing room? If it's accepted, it definitely gives a precedent for later agreements concerning Galicia, as well as minorities within the Kingdom of Hungary. It could cause issues with the newly united Germany, but given the time frame I'd doubt they'd be
seriously angry or try to invade or anything like that.
In the long run, it'd give the Austrian portion of the monarchy an ally against the Magyars, as they would effectively be able to twist Hungary's arm if they proved resistive. In the short term though, it could be unstable... but not anymore unstable than politics in Austria-Hungary were in this period anyways, what with Czechs boycotting the Austrian Parliament for nearly a decade, unworkable governments, and weak Prime Ministers who had to deal with a lack of majorities in Parliament and had to form overreaching coalitions (like Taaffe and his
Iron Ring of Conservatives, Clerics, and Slav allies).