The newspaper which Mike is carrying on his way to his first shift guarding Claire has the word "SUPERMEN!" on the back page. He is carrying an identical newspaper three or four days later.
Claire is allowed to stay in her mansion, despite the fact that she is a witness in a high profile case and this wouldn't happen. They would have likely moved her to a secure location were no one could find her, and should have multiple cops around her at any given time, so the plot of the film makes little sense.
Claire and Mike are seen frequently going out and doing things like drinking and going to parties. This would not happen in reality, as Claire would be in witness protection and she wouldn't be able to do this. Doing this would cause the police to disown her because she is not following police protocol, and this would be considered abandoning her government contract and they would leave her to her fate, this just wouldn't happen in reality.
Why is the NYPD watching what would be considered a federal witness? They may have done this for a couple of days, but not the extended period seen in the film. This would have likely been a FBI operation after a couple of days, and they simply wouldn't take a NYPD officer off the streets this long. It just wouldn't happen.