Two possibilities:
First, as others have noted, the situation in the balkans will have to change so that we get some kind of renewed Crimea coalition of the UK, France, Italy and the Ottomans.
Now, the OTL development of 1876 to 1878 left the Russians angry at A-H for spoiluing the Russian plans and at Germany for Bismarck letting it happen. Can we get an situation where Vienne is supportive of the treaty of San Stefano and mainly London opposes it? Probably not. Unless Russia commits firmly to Austrian goals beyond SE Europe., like, hm, helping Austria conquer Italy. But that is silly.
So we might have to derail it earlier - by making the 1877/78 war a crushing defeat for Russia, perhaps accompanied by cossacks commiting some atrocities that turn Western European opinion against Russia. So, the different Dabubian principalities stay Ottoman vassals, a somewhat stronger OE and the UK begin to patrol the Black Sea, France and Italy tag along. The Russian-Austrian rivalry is postponed, since the Sick Man doesn't look so sick in this TL.
But what is in this for Germany? Bismarck wanted the 3EL since it would help to isolated France, his main goal. The whole construction above does not help that much, since France isn't interested in the whole Ottoman situation that much by 1880 - well, in the Levant, but that agains is not Russias main interest.
Another possibiliy, much closer to home: another Polish insurrection during the 1880s. Keeping the Polish people down was a traditional pastime of Prussians, Russians and Austrians. A violent Polish uprising, perhaps with ties to secret French support, would help tie the 3 empires closer together and turn them against France. For that, we would need a much more adventurous French government.