Hopefully Not a Stupid Question: Flash vs. Polarizing Filter

Quick question that Google is not helping with... Will a circular polarizing filter have any impact on getting correct exposure when using on-board flash? More specifically, what about when the flash is being used outdoors for "fill"? ....

Most of the polarizers (if not all of them) do slightly reduce the incoming amount of light, so if you're shooting "old school all-manual mode", WITHOUT the flash, consulting your external lightmeter, you got to compensate for it. With the old flashes, manual or auto mode (where you manually choose ISO/ASA, F stop and the sinc speed), you've got to compensate. Now, with the flash being set to TTL mode, the exposure system should do it for you (and the built in flash is TTL compliant, the lens has to be compliant too, of course). As for the "fill mode", you should compensate too, I believe.

Flash is directional light. Polarizing filters are directional. Using them in conjunction means that the polarizer will negate the flash's impact if and only if, it is casting light that the polarizer is negating, which I don't imagine it would if you're looking to darken bright skies.

Best advice I'd heard (for fill situations), and something that's stuck with me, is that you expose for the background light you want, and use flash to augment that on top of it. If you're using a polarizer for skies and whatnot, and a flash to illuminate the foreground/subject, I don't believe you'll have an issue.

Yes, flash gives directional light, but, you cannot precisely estimate the direction of REFLECTED light, casting shadows around... Once the flash has fired, depending on the object's distance, shape and other qualitative/quantitative elements, you'll get a pretty weird mixture (of light) in terms of direction. And if the surroundings are made of glass, water, haze/fog...?
But, anyway, a little test would reveal the secret for you - try to manually compensate for as much as +1/2 of F-stop (a half), and see if the exposure is correct...

I'm thinking this is going to require first hand experimentation...

....

When I do product photography of shiny objects, a polarizer with flash is absolutely necessary otherwise I'll spend a lot of photoshopping time to get rid of hot spots. Yes, the polarizer will take down the exposure by 1 to 1.5 stops but it does save me time post-production. Attached is a typical example - without the polarizer, the metallic parts as well as the corners of the polished leather case will be over-exposed requiring post-production dodging.

And yes, with circular polarizers, it's less predictable so it's trial and error to get polarization angle right the first time. But once the positions of the speed lights are set, I just have to make sure the polarizer don't change position. The good thing about circular polarizers also it that they are less sensitive in angular position than the old linear polarizer days. I did absolutely no post-production dodging on the attached photo - thus the power of the polarizer!