President Obama -- Cuban Cigar Crisis ... La Foto Es Verdad

Move over Monica ... there's a new controversial cigar in the White House -- a Cuban fingered by President Obama in a photo many thought was fake, but we've learned it's the real deal.

The original pic shows Obama Wednesday night ... sampling the aroma of what's alleged to be a Cuban cigar. It was posted on Reddit Thursday morning, and immediately the Internet started tearing it apart as a Photoshop job.

TMZ has obtained several high-res pics of POTUS' run-in with the cigar -- and our photo/art experts say everything about them looks legit. Significant ... since Cuba and the U.S. just agreed to open embassies in each other's countries.

According to our D.C. sources ... the hand-off went down during the White House Hanukkah party -- when an invitee walked up and said, "Mr. President, here's a Cuban." Obama proceeded to give it a huge sniff and then held on to the gift. It's unclear if he pocketed it or handed it to the Secret Service.