If you wish to advertise a position anonymously it is entirely possible by just having everything sent to an email that doesn't disclose the name of your company.
If you apply to an anonymous ad, it means you will not have your identity disclosed without your prior consent.
If you apply to an ad that came from you own company, nobody but me will know, because a clear and strict rule is, in such a case, that neither you nor the company will know. You would get the same answer as other people "your profile doesn't match the client's requirements".
If you apply and your resume shows some kind of link with the client you will be asked for further details in a way you couldn't understand what I am checking.
I, as former executive in The Bahamas as well as in France will guarantee that your personal information will never be disclosed without your prior consent. It means also that any information you provide could be double checked and any falsehood will result to your resume being disqualified from the list provided to clients. Good luck in your search, Thierry Boeuf ps: beside my professional background I wish to add that The French government trust me enough to represent it in The Bahamas and preserve its confidential information as Honorary Consul.