How about this: the French effort to gain control of Mexico in the 1860s succeeds and the Franco-Prussian War is either avoided or ends in France's favor, allowing the Second Empire to continue. Germany still unifies, though.
Buoyed by its success in Mexico, France starts throwing its weight around in the Caribbean and Latin America just as it is in Africa and Asia. It secures an alliance with Russia, Austria-Hungary, and Italy to counter the threat of Germany in Europe and Britain in the imperial game, thus driving those two countries together.
The United States, fearful of the French threat right on their doorstep, abandon their policy of isolationism and embrace the Anglo-German alliance.
NOTE: The POD required for this is a pre-1900 one.