Hmm, I've been drinking pine tar extract (as cough medicine) and now I'm feeling a bit peculiar and smelling a bit like turpentine, so anyway I'm just going to maybe update this thread a bit until I'm not feeling so dizzy and weird and whatnot.
So, Fortean anime, that is Fortean, hmm.
The Melancholy of Haruhi Suzumiya: (This is really quite Fortean.) The story of a well-meaning but annoying Japanese high school girl (Haruhi), who is unwittingly omnipotent. That is to say, she basically has "Q Continuum" abilities, but she doesn't realize it. Her whims and moods and daydreams can alter the fabric of reality. She must never realize she has this power, because if she did, it would be... very not good. An alien super-intelligence is monitoring her activities, and trying to make sure that she doesn't get too many loopy ideas. Unfortunately Haruhi is obsessed with strange phenomena, which is a problem. A time-traveler and and esper also show up, both also striving to prevent Haruhi from thinking about anything too wacky. Haruhi's mopey and dickless sort-of boyfriend is similarly tasked with this task. He is the narrator, and he mopes a lot. It is pretty much okay.
Spice and Wolf: The story of an amoral sociopathic peddler, and his dipsomaniac shape-shifting wolf-goddess girlfriend. Set in a sort-of Europe during the sort-of Dark Ages. Problems arise because the peddler is not a total bastard, in spite of his greedy, sneaky little prick persona it turns out that deep down he is really a nice guy, which sort of pisses him off, and he frequently is conflicted between profit motive and doing the right thing, and also his girlfriend just wants to get drunk off her ass all the time, and vomit and eat apples and vomit some more. Also neither of them want to acknowledge their feelings for each other, and so each day is a charade of petty bickering and shenanigans, and each night is a purgatory of silent yearning and loud puking. All in all it is very wistful and poignant, you will probably like it.
Sora No Woto (Sound of the Sky): Set in a distant future, on a quite possibly doomed Earth. The world's ecosystem is in decline, and civilization is gradually fading. War may be impending, naive young recruits do their best to stay true to themselves, there is innocence and also lots of poignant wistful stuff, you will probably like it.
Watamote: The story of a high school girl who suffers from extremely severe social anxiety disorder. Very dark comedy, hilarious and disturbing at the same time, poignant, wistful, you will probably like it.