Federal prosecutors have dropped their charges against a homeless man accused of torching an unmarked NYPD car in Brooklyn — after his defense attorney said they had the wrong guy following a “half-baked investigation.”

Michael Rodriguez, 32, was freed Monday after a federal judge officials dismissed the charge against him. He could have faced 20 years in prison if convicted.

“I’m pretty sure they had the wrong guy,” public defender Kannan Sundaram told The Post on Tuesday.

“I’m not sure what made them realize it. It sort of smelled from the beginning. It was a half-baked investigation.”

Prosecutors did not give a reason for seeking to dismiss the charge against Rodriguez. A spokesman for the U.S. Attorney’s office in Brooklyn declined to comment.

The charge was dismissed “without prejudice,” meaning prosecutors may re-file the case if new evidence emerges.

Rodriguez was jailed Thursday after the federal Bureau of Alcohol, Tobacco, Firearms and Explosives accused him of methodically setting fire to a sedan around 4:30 a.m. on June 2. The car had an NYPD placard on its dashboard and was parked in Williamsburg.

Charging documents alleged that Rodriguez was caught on security camera footage pouring an unknown liquid on the windshield of the vehicle before placing cardboard over the liquid and lighting it on fire with a lighter.

Prosecutors said he then stayed at the scene until the car was fully engulfed in flames, taking photos or videos on a cellphone.

Authorities had released a photograph of the suspect and visited a “public area” where Rodriguez had been known to sleep, court filings state.

A tipster told investigators that a man known as “Justin,” who had been sleeping in the area for two weeks, had asked whether the tipster “saw anything about him in the newspaper” the day after the surveillance photo was released.

A newspaper article about the incident was found near where Justin had been sleeping.

Investigators approached a man three blocks away who “responded to the name Justin but later admitted his name was Michael Rodriguez,” according to the criminal complaint.

Rodriguez’s defense attorney claimed that, at the time of his client’s arrest, authorities had not yet searched the cellphone, which they recovered from a location they were led to by the tipster.

“My sense is they were able to search the phone and it did not match up to my client,” Sundaram said.

“The location where they found that phone is not where my client stays anyway,” he added.