Hetalia fandom is full of works regarding current and past politics and unions. This is (obviously) pretty much only feasible in this continuity, considering that the show is about personified countries.
The "Letters" trend: characters answering fanmail being sent to them (via reviews by readers). The strange thing about this is the fact that A) these typically don't require/have an actual plot, and B) interactive fanfic is actually against the rules on FF.net, but they manage to make up a VERY prevalent chunk of the APH category there. Occasionally sloppy research shows up, since some feature countries/cities/places that the authors do not hail from/have been to.
There's also a developing trend in the form of "blogs": in others words: a country is forced to/secretly wants to make a blog and in the same vein as the letters fanfictions, answers the comments "posted" to the character's blog while at the same time providing an excerpt about that particular character's day or current thoughts, much like an actual blog.
An APH fanfic writer's motto could practically be "If you look hard enough, any pairing can be historical." And it's generally true.
The APH fandom is infamous for its Mpreg. Any geographic or administrative region could potentially have its own personification, and for some fanfic writers, there's no better way to Hand Wave in an OC than by having two characters go to town. Search in the right place and you will find Mpreg for any and every Hetalian.
AU fanfics with the characters rewritten as ordinary people instead of countries are especially common, since they allow the author to write certain pairings/situations without any of the political/historical stickiness.
AU fics in general are pretty common, normally coming in the form of Alternate History, For Want of a Nail and/or Dark Fic. Seeing how the characters (and their relationships/sanity) would be like in the future or had events played out differently from real life does allow writers certain freedoms with research.
The stereotypical Gerita story. One day, Germany finally realises he feels romantic feelings for Italy, tries to woo him, fails because Italy says "I'm waiting for someone" (HRE). Germany angsts until Prussia/Hungary/Austria/Whoever tells him he is Holy Rome! He tells Ita-chan, and they live happily ever after.
Even though, in series, Prussia continues to exist past his country's dissolution, many fanfics go the vein of Prussia dying and his friends and family mourning him. In addition, there's a line of fanfiction in which there is only allowed to be one representative for Italy, and one Vargas brother (usually Romano) must disappear.
The Globe Fic: A human gets an item (usually a globe of some sort), that summons the Nations into her (usually it's a she) life. Hijinks ensue, with the human often playing the part of the Only Sane Man.
A common plot, which is normally exclusive from the Latin American fandom and also overlaps with Sailor Earth, is adding nation-people from other Latin American and Spanish-speaking countries, besides Spain and Cuba. Some of them interact with the canon cast and sometimes their plots are completely independent from the official canon. Most of the times those nation-people are depicted based in some well-known stereotype Latin Americans has about themselves or their neighbors, being the most common one is the personification of Venezuela being depicted as a Hugo Chavez (or Nicolas Maduro) wannabe. Sometimes, they went farther and some fanfics even ignore Cuba's canon personality just to depict him as a Fidel Castro-wannabe.