The hate mongers behind the deadly Jersey City shooting that left three civilians and a police officer dead were a pair of religious-fanatic lovers who lived together in a van, according to records and neighbors.

The killer couple, David N. Anderson, 47, and Francine Graham, 50 — who died in a gun battle with cops — met about two years ago when Graham was living a quiet life as a home health aide in an Elizabeth, NJ, apartment complex, The Post has learned.

“She was desperate for a boyfriend,” said neighbor Kenneth Safford, 69.

“She smoked cigarettes out here. We would talk. She always said she had a problem … She couldn’t get a man,” he said, adding that she sometimes speculated as to why she couldn’t find love.

“She claimed she had built-in body odor,” he said. “She wanted a boyfriend so bad.”

A second neighbor — a 45-year-old woman who did not want their name given — said that after meeting Anderson, Graham changed.

“Since he got into her life, she didn’t have a desire to work. She used to exercise with me, we used to walk around the block. Then all of the sudden she seemed to disappear. I stopped seeing her around. She stopped taking my calls,” she said.

Graham also told the woman she was part of a religion pertaining to “black Jews” — likely the Black Hebrew Israelite movement, of which Anderson has been identified as a one-time follower by law enforcement sources.

The group believes they are the true descendants of the ancient Israelites, are known to vilify white and Jewish people and are considered a black supremacist group by the Southern Poverty Law Center.

Anderson and Graham were heard chanting and reading the New Testament from their home, another neighbor told The Post.

“She used to do chanting and stuff like that. It was loud enough to know it was that. She would just be standing right there,” the neighbor said. “Standing and smoking. You would just hear them just chanting religious stuff.”

About a year after the two met, Graham — who grew up in the Manhattanville Houses in West Harlem — lost her home through foreclosure and moved into a van with Anderson, the neighbor said.

“They would come back to the apartment once in a while and stay until the sheriff put the lock on the door,” the neighbor went on.

Another local, who only gave her name as Alexis, said Graham “was a little off” and “yelled at the kids” in the neighborhood for playing in the street.

Before Anderson met Graham, he was in the Army Reserve, serving as a fuel and electrical system repairer as a specialist from September 1999 to September 2003, according to an Army spokesman.

But shortly before he got out of the reserves, he was arrested for having an illegal weapon and went on to spend the next eight years in and out of jail in New Jersey and Ohio, records show.

In 2003, Anderson pleaded guilty to unlawful possession of an air/spring pistol in Hudson County, NJ. He was sentenced to three years’ probation in February 2004, having already spent 335 days in the county jail, court records show.

Just over three years later in November 2007, Anderson was found guilty yet again of the same charge and was sentenced to probation again in December 2008 after spending 445 days in jail, the records show.

Once he was sprung, Anderson moved briefly to New York and then headed to Kent, Ohio, in July 2009 to shack up with a woman he’d met online, according to records from the Kent Police Department.

On Sept. 20 that year, the two got into a fight about having children together, leading Anderson to punch a hole in the wall and threaten her.

“I’m gonna kill you. I feel like killing you. You made me lose everything, you can leave this world. Call the police because I’m gonna kill you,” Anderson allegedly told the woman as he forcefully grabbed her cheeks on her face, the police records state.

He was arrested for domestic violence threats and later pleaded the case down to criminal mischief and spent 30 days in jail, Ohio court records show.

The couple got back together after the quarrel and about two and a half years later in April 2011, Anderson was arrested again when police realized he had an open warrant from Hudson County for his 2007 weapons violation during another incident involving Anderson’s girlfriend and her children.

He was extradited to New Jersey and was sentenced to five years in prison in June 2011 for violating his earlier probation — but only served four months in prison before he was let out on parole again.

Anderson and Graham are accused of walking into the Jersey City Kosher Supermarket on Tuesday afternoon and shooting three individuals after they had killed a Jersey City detective in a nearby cemetery.

The attack is being called a hate crime by Jersey City Mayor Steven Fulop, but Jersey City police have yet to say if the shooting is being investigated as a hate crime.

The duo are also considered the prime suspects in the murder of a Bayonne Uber driver.