Are there any?

Many agencies now require a photo of a drivers license and details about where you are employed to become a patron.

What realistically happens though if they get in trouble for tax evasion etc?

Forget the potential embarrassment of being on a client list, what happens to you purely from a law standpoint? Can you be charged with a crime merely for being an approved client on their computer with a record of past appointments?

My understanding has previously been that unless you are in a hotel room recorded uttering the boneheaded phrase “sex for money” in some form, you can’t really be charged with anything that would stick.

Is this an imperfect understanding or mostly right?

I hope someone more versed in the law than I will discourse fully on this matter, but ...  
"you can’t really be charged with anything that would stick" isn't of much comfort because you can be charged with things that won't "stick" but will put your name on more embarrassingly public lists.  Escaping conviction in court might be cold comfort after those who knew you all flee in disgust.  
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This is only a small part of an answer, I know.

“Escaping conviction in court might be cold comfort after those who knew you all flee in disgust.”

I’m fortunate to work in an industry where it’s more probable I’d get teased or high-fived by coworkers or a boss than fired if it came to light that I had seen a “lady of the evening.”

I can imagine it wouldn’t go nearly as well for other people however, and I’d rather not find out regardless.

From my understanding........Unless there's proof on what the Client says to the Provider, then the Provider is more in trouble then the Client.