The first half is appalling. It's a grating, obnoxious mess, as if the producers had taken a bet to see if they could drive away all the viewers. Later it improves, but... sheesh.
Firstly, the show's premise. This is insanely complicated and a train wreck. It's adapted from what are technically a series of light novels, except that they're not so light. These thirteen episodes adapt its first two volumes, of which the second is 1152 pages long. Furthermore, their author (Minoru Kawakami) has set them in a fictional universe of his creation that's split into six eras of future history, explored in dozens of novels and containing a ridiculous number of ideas, characters, species, settings, political systems, supernatural powers, etc. (Horizon in the Middle of Nowhere is set during his universe's GENESIS era, which is the fourth in chronological order.)
If you make the mistake of buying this show on DVD, start by going to the extras of discs 2 and 4. Watch the options called "Far East's Enlightening Lecture". These are basically big info-dumps and you'll need them. After that, bear in mind that the indigestible info-dumping in the show's first couple of episodes is: (a) essential, and (b) insufficient. Pay close attention and try to memorise it all.
This won't stop the show being an ugly mess, but at least you'll have given yourself a fighting chance of managing to make it through to where the writing starts showing signs of life. I'll try to help here too, though.
Long ago, mankind abandoned the trashed Earth and fled into outer space. This didn't work. Some unknown phenomenon was preventing interstellar travel. Mankind thus had no choice but to return and live in the planet's only remaining habitable area, i.e. Japan, plus some pocket dimensions they built to house the excess population.
They then started reenacting human history according to the Holy Book Testament, which they think will help them return to outer space. Don't ask me why. I don't get it either. So far they've got up to 1648 and they're about to run out of the Testament's historical predictions, which probably means an imminent apocalypse. Anyway, the world's many current geopolitical powers include the Catholic Church, Italy, Spain and the city-ship of Musashi, which is home to the original population of Japan that got booted out when the pocket dimensions invaded in 1413. These countries' governments combine historical reenactment and utter barking lunacy, so for instance everyone in Musashi's government can't be older than 18. That's the law. Once you've left school, you're no longer allowed to rule the country. Is this the epitome of ridiculously powerful student councils in anime? All these countries regularly go to war with each other in ways that are theoretically a bit like what's in the history books, but in practice are mostly about stabbing each other in the back and so on. I might be wrong there, though. I'm still a bit vague on that one.
Meanwhile the world's inhabitants include humans, elves, giants, spiritual races (e.g. dryads, fairies, werewolves), gods and archdemons, angels, robots with souls (e.g. automated ninja super-maids) and other mechanical beasts, dragons, talking red gelatinous blobs, various forms of undead and support device races (e.g. "mice", which are actually a bit like a witch's familiar). There are also giant piloted war robots, like Gundam. There are also superpowers, which I think are based on weird technology but tend to behave in mythological or fantastical ways. I could go on and on.
So you've got a stupid amount of detailed weird SF background, which you almost certainly won't take on board, in a storyline that's too abbreviated and choppy to make much sense. That said, though, the second half of the series improves on reaching a story arc that even this anime can't ruin. (It's still going too fast, though. The Toori/Kimi childhood flashback scene in ep.9 is good, for instance, but rushed.)
Then we have the things that are actively obnoxious. As well as being a space epic of historical scale, it's also silly exploitative otaku-pandering trash.
There's the law about everyone in Musashi's government being under 18, for instance. That's just... what? It makes my head reel. Then we have the character designs. Look at the DVD covers. It's the kind of thing you'd only want to buy in brown paper bags, not because it's sexually explicit (it's not) but because it's embarrassing. You could land a NASA space rocket on those women's breasts. Even skyscraper-sized war robots have boobs. In fairness, though, it's of a piece with all the other design decisions in this show, in which a character's outfit is liable to make them look like a walking cathedral.
Google "Nate Argente Loup Mitotsudaira", for instance. Look at that hair. How can you not love a show mad enough to do something like that? Tomoko thinks this show is unmitigated garbage, but she's in love with Nate's hair. It comes down from her head in four purple coils, each one much bigger than her body. She looks like a jet engine. Tomoko called her "the drill". That might conceivably be the stupidest hair in anime, which would be an achievement akin to "heaviest black hole in the universe".
Then there's the show's protagonist, Toori Aoi. His main character trait in the show's first half is fondling women's boobs. That's his favourite subject. Well, that and pornographic video games. He'll grope his fellow students in class and no one asks him to stop or even finds it offensive. On top of that, though, Toori has the most punchable face I've ever seen. You'll be reaching for a frying pan. He's got these idiot droopy eyes and a big idiot grin... ALWAYS. His expression never changes. He'll always have the same smug vacant half-asleep look while groping women, talking amiably about groping women or even having an entire scene that's building up to the comedy groping of a woman. Everyone likes him and worries about him, but that's simply because he's the light novel protagonist and so that goes without saying. There's no visible reason for it. I wanted a piano to fall on him.
However he's not the biggest pervert in his class, because there's also a fat comedy paedophile. This man denies being a paedophile and instead claims to be something else that sounds almost the same. Oh, and there's also a racist stereotype Indian who carries curry on his head.
And the plot appears to be mostly built around fights. Big fights. Action scenes, wheeee. There isn't always any reason for them, though, e.g. Toori's teacher challenging her class to a big fight in ep.1 simply because the production staff want ep.1 to look exciting. The plot is impenetrable because we have scenes like that instead.
Tomoko was startled to see that this anime is by Sunrise, who have a strong reputation. They're famous for mecha shows. "What happened?" said Tomoko.
It's like watching a car crash. It's a laundry list of unwatchable anime flaws. The only thing in its favour is that it's theoretically family-friendly, in that it doesn't have on-screen sex or nudity. Personally though I can't say I'd recommend Toori and his merry chums as any kind of relationship role model.
However there are also things I like. You'll have to grit your teeth to make it through to most of them, though.
The show contains some memorable ideas for characters. I don't think the characters themselves are memorable, because they're insufficiently developed. Not enough screen time. The show goes too fast and tries to do too much. However I liked the roommate in the wheelchair, or the girl who was forced to have half a sex change by her parents for the sake of inheriting a title.
The plot comes alive in the second half. Toori does some things that are cool and clever, although the cleverest thing he does is a workaround for the fact that he's an idiot. This world's definition of "fighting" turns out to include debates, which are actually quite good. I enjoyed them. I think the show wants to be incisively exploring logic and politics, like a Mamare Touno show (Maoyu, Log Horizon), although unfortunately it's got way too much magic nonsense for that. Mamare Touno is genuinely intelligent. This show is more like something that would like to be intelligent and sometimes sort of gets there. (Tomoko called it chuunibyou, in that it likes using long words and jargon it's made up to make what it's saying sound more impressive.)
Stuff happens. An innocent girl gets sentenced to suicide for political reasons. Countries go to war. The Pope turns out to be a world-conquering bastard, not to mention funny in his pettiness over winning a non-battle in ep.11. Soon afterwards, there's a bragging match between two other characters that's just as amusing.
I should also admit that I quite liked Toori's goal of confessing his love to his childhood friend who died ten years ago. That's memorable, at least. Some tragic backstory never hurt anyone, although Toori is tragedy-resistant.
I was dumbstruck by the first two episodes, which I'm not sure I'd even call watchable. I then merely disliked the next two and thought there was something worth watching in eps.5 and eps.7-13. The show never stops flailing all over the place, with that combination of kitsch, sleaze and action-orientated political melodrama, but it seems possible that it's now found its feet. Season 2 might be good. I wouldn't put money on it, but it's conceivable. I like the debates. I like some of the characters. It can be dramatic, but then it'll have a boob-groping battle. Similarly the fights at their best can be awesome, but at the end of the day I'd prefer the show with less fighting.
However I'll always have a (small) soft spot for any show with a girl whose weapon is a spanner that could take the legs off an AT-AT Walker. The show's still all kinds of wrong, though.
"The knights still stand superior over the commoners!"