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Unless you work for the government then your constitutional rights are not implicated by the facts in your post. This is so because, subject to very limited exception, the constitution only provides rights with respect to action by the government.
Unfortunately, the general rule in private sector employment is that employment relationships are "at will" absent an express agreement to the contrary. At will employment can be terminated for any reason not amounting to discrimination on the basis of a legally protected trait (race, religion, gender, etc.) or retaliation for engaging in certain forms of legally protected conduct (filing a wage claim, taking FMLA leave, etc.). It doesn't matter whether the basis for termination is fair, reasonable or even TRUE.
Thus, it would not be illegal for your employer to take adverse employment action against you on the basis of anonymous accusations--even if you could conclusively prove those allegations to be false.
The thing is, its serves no purpose for an employer to fire an employee without a good reason. Firing employees costs an employer money in the form of increased unemployment insurance premiums, plus advertising and training expenses for the replacement. Therefore, while employers are not legally obligated to be fair, they tend to be because it is in their own self interest.
So, all you can really do here is attempt to reason with your employer that you believe the anonymous allegations against you are unfounded and personally motivated based on disagreements you have had with your suboordinates. Any reasonable employer would give that explanation some serious consideration. But, at the end of the day, you cannot force your employer to be fair, and your only power is the power of persuasion.
I hope that you find this information helpful. Please do not hesitate to let me know if you have any questions or concerns regarding the above and I will be more than happy to assist you further.
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