On the surface, this is a courtroom drama. You'd probably file it next to 12 Angry Men and just move on, plus, they came out the same year.   
  
But if you actually think about it, their philosophies are total opposites. 12 Angry Men is basically this comforting assertion of truth, the idea of: what happens if we put twelve guys in a room and they talk honestly long enough, will they somehow arrive at the pure, objective truth and the system works? And we all go home happy? Or what?

Witness for the Prosecution is game is game at it's purist form: the most skilled performer wins, not necessarily the most truthful person.

Sir Wilfrid is the most brilliant legal mind in London, sure, but he's really just a high level player driven by his own ego. The entire film is a fog of war, and the trial is just the battlefield.

Usually, for something to be noir, you need certain ingredients:

Shadows? Check.

Cynics? More than check.

Trench coats? Check one time.

Femme fatale? What more could you ask for than Dietrich?

Marlene Dietrich as Christine is just the classic noir archetype, but premium edition.

She's the woman who knows too much, except the entire film is structured so you can't even tell what she knows or when she's lying about knowing it.

But there's no traditional detective here.

The guy is already caught before the movie even starts, so what's there left to detect?

The cross examination IS the detective work!

It's just been relocated into a courtroom. Christine doesn't need lust or a gun to manipulate people.

(spoilers!)

She runs a flawless, god tier psyop. She realizes that if she simply acts as a loving wife giving an alibi, the jury is going to discount her.

So she plays the cold, calculating, hateful wife testifying against him, setting herself up to be destroyed by Wilfrid on the stand.

She weaponizes reverse psychology against Wilfrid’s brilliance.

And that brings us to the real detective, which is you.   
  
You, the viewer, have basically the exact same information as everyone in that courtroom. There are no wait for the twist moments, nor any hidden information or secret plot like we see in The Big Sleep.   
  
So you're running the trial in your head the entire time. You're trying to read micro expressions, weighing the odds, auditing your own belief live. What would I argue here? What's the angle? Do I believe this person or not? Why?   
  
So Marlowe, how are you going to solve this?

But let's be honest about what your actual role is here.   
  
You're not Marlowe.   
  
You have the exact same skin in the game as the jury, which is absolutely zero.   
  
Think about it. You're asked to sit down for a couple of hours, exactly like a real jury. You don't know these people. It's not your kid on the stand, it's not your partner. There's no real upside or downside for you.   
  
If you get it wrong, you don't lose any sleep really, you don't get shot, you just turn off the film and go by your day. They just go home and forget in two days.  
  
Because you have no real skin in the game, how much actual mental compute are you really going to dedicate to this? Are you really going to squeeze those neurons for two plus hours to rigorously audit every single inconsistency, calculate every probability, and stress test every variable?? Hell no.  
  
It costs too much energy, and there's no payoff, the ROI just doesn't make sense.   
  
You just sit back, take the path of least resistance.  
  
Christine knows this, and that's how she runs her exploit.   
  
She knows that a room full of tourists won't dig past the surface if the surface performance is good enough.   
  
You literally are the jury. You let a criminal walk away.

Everything up to this point is game theory optimal play from everyone involved, (or at least close).  
  
BUT:

This film would've been too dark if it wasn't for the Hays Code/BBFC.

The only flaw in the film is the final couple of minutes: The stabbing at the end doesn't really serve any purpose except for the Hays Code.

So you're telling me, this master manipulator who just beat a murder charge would instantly break his opsec and flaunt his new girl in front of this 4D chess maniac wife who literally holds his life in her hands? And this wife somehow has a knife in a courtroom and stabs him? What?

It's just Billy Wilder paying the censorship tax so the film could actually get released.

Tyrone is here, got me thinking about Nighmare Alley.

William Lindsay Gresham's book is much, much more...dark.

"Mister, I was made for it..."  
  
Very, very good film.

Witness for the Prosecution (1957): The noir version of 12 Angry Men. YOU are the detective!