Read it to critique it, ended up upset that I didn't read this before due to my general lack of attraction towards AU stories. Very enjoyable, does a lot of things right. Characters and interactions are fun to read and distinctive, moves at a good clip, prose flows nicely. Kind of reminds me of one of those Dresden Files books, and those are massive commercial successes so you know that's a good thing.
Without it being finished I can't offer many meaningful comments on the plot but it kept my interest and isn't predictable (to me, anyway) and that's all one can ask for. Really I only have one major complaint and a few nitpicks, and those are out of the realm of "this is objectively bad and you should change it" and into "I really think changing this would have improved the story, but in the end it's debatable"
It's not something that can really be 'fixed' at this point but there's one thing I think would have turned it from an enjoyable read to a real 'page turner' as you'd call it. A story where you're constantly eager to find out what happens next and f5 the thread several times a day, and it's nothing complex or crazy. It's the basic storytelling elements of a clear looming threat and clear antagonist.
First up is a real antagonist. Say, seeing the Tenma and/or some kind of tengu figurehead do clearly evil stuff to the people of the city two or three times, darth vader style, would really fuel that feeling of righteous anger in the reader. Right now they're a vague background threat that we're told are bad. Sure the main character gets attacked by one of them, once, but something a little beyond that would be good, I think. It focuses the story. I know there's Kotohime, but again she's kind of nebulous in her capabilities and threat level on top of being some kind of spirit/thought-form/whatever and we're not 100% sure what her deal is.
Same thing with a looming threat, a real stake, a ticking clock. This will probably happen harder later in the story, but until now it hasn't been there. Somewhere around the end of the first act is where this would usually surface, then the stakes would be raised again later: rather than just sleuthing at their own pace and dealing with trouble as they come to it, something happens that forces them into action at first, then into urgent action later!
Suwako realizes some kind of magical thingamabob Kanako used to bind her is slowly corroding away her power for unknown nefarious purposes, and the effect might be fatal! Boom, instantly there's tension, motivation, drama.
Oh no, the Tengu are kicking their search into overdrive for some reason, kicking doors down in the slums and spreading the word about the fugitives - with rewards attached! Suddenly they can't calmly walk the streets anymore!
Of course these are just silly examples, but you get what I mean. Unless this turns out to be a very, very long story, fantasy epic style, and we're early in it, I'd expect to be seeing some of those elements pop up by now. Instead the stakes and driving forces we had were: finding Shizuha to pay for a mild debt. Investigate a soup kitchen because it's suspicious. Help the goddesses raise faith because we like them and the way of the tengu is worse in some vague way that we don't have concrete examples of.
I'm a fan of structure in stories, because it's tried and true for a very good reason. This story is the story of a private investigator with a dark past in a scummy city. There's action, there's romance, there's adventure, there's banter. I'm approaching this critique with the mindset that your goal was this genre, a fun adventure romp in the style of the good old commercial novel. And I'd say it's successful, but lacking that final push to become a real masterpiece. The lack of those elements does result in a story feeling slightly more relaxed and opening up space for more exploration of other areas, and if that's the goal, fantastic. But reading it, I felt like it would have been truly exciting and unique for THP with the action push, more like a traditional detective-fiction-adventure thing, and it would have fit right in with all the other elements for a story worth of a real publisher, fanfiction aside.
But that's why it's just, like, my opinion, man.
Still, I'd rank this up there and I'll definitely keep reading in the future. If you got this far, thanks for reading my rant. I didn't edit it too much.