You know within about two seconds of picking one up off the rail. A proper wool coat has a bit of heft to it, enough that it pulls slightly on your arm when you lift it, and the surface has that faint roughness that's nothing like the smooth, slightly plasticky feel of the blends sitting next to it. Run your hand down the front and it shouldn't feel floppy or want to crease under your fingers. It should feel like it could stand up on its own if you propped it against a chair, which sounds daft until you've felt the difference next to one that's mostly polyester wearing a wool label for confidence.
Sizing is where people come unstuck, because a wool coat has to go over jumpers and sometimes another cardigan under that, so trying it on with just a t shirt tells you nothing useful. Put it on over whatever you'd actually wear underneath in January and check the shoulders first, because that seam sits where it sits forever, there's no taking it in later without a tailor and a conversation about cost. If the shoulder seam drops past the edge of your actual shoulder you'll look like you're wearing someone else's coat even if everything else fits. Sleeves should let you bend your arms and still cover your wrist bone, check this by actually reaching forward like you're grabbing a pram handle or a bag off a hook, not just standing still in the mirror. The other thing nobody checks is whether it closes properly over a bit of midriff thickness without gaping at the button, because a coat that only looks good buttoned up on a thin day isn't one you'll reach for in a real cold snap. If you've only got room for one, go for a plain colour somewhere between camel and a dark grey, knee length rather than long, with a collar that stands up round your neck rather than lying flat, because that's the version that doesn't fight with anything else you own.
Underneath, this is the coat that makes a jumper and jeans look like you tried, and it sits just as well over a dress for something slightly more done up, which is most of what a normal week actually needs. It doesn't ask you to match it to anything, it just sits on top and looks like the grown up in the outfit.
Next time you're looking at one, go straight for the weight in your hands before you even check the price tag.