EXCLUSIVE

A transgender woman is going to war with CrossFit -- claiming the fitness titans refused to let her compete in the women's division at the upcoming CrossFit Games because she was born with a penis ... TMZ Sports has learned.

Chloie Jonnson has filed a $2.5 MILLION lawsuit against CF -- claiming she underwent sex reassignment surgery in 2006 and is legally recognized as a woman by the State of California.

But when Chloie -- 5'4", 151 lbs -- asked the powers that be about competing in the Games with the ladies, she was shot down hard.

In fact, TMZ Sports obtained a letter from CrossFit to Chloie in which the CF pulled no punches -- "We have simply ruled that based upon [Chloie] being born as a male, she will need to compete in the Men's Division."

CF argues ... "The fundamental, ineluctable fact is that a male competitor who has a sex reassignment procedure still has a genetic makeup that confers a physical and physiological advantage over women."

But in the letter, CF gets aggressive -- saying, "Our decision has nothing to do with 'ignorance' or being bigots -- it has to do with a very real understanding of the human genome, of fundamental biology, that you are either intentionally ignoring or missed in high school."

Boom.

Chloie says CF is full of crap -- claiming the company is violating her civil rights by not letting her compete as a woman ... and she wants at least $2.5 million to make things right.