CIA Schools Katherine Heigl's TV Show on How Prez Does Business

The CIA is watching Katherine Heigl ... but she's not in danger.

Turns out the super-secret agency is fixated on Katherine's new show, "State of Affairs" ... to the point of challenging the authenticity of the program and even live tweeting the way Washington really operates.

A CIA spokesperson tells TMZ ... "State of Affairs" helps the agency "counter popular myths and misconceptions about the Agency ..." The spokesperson adds social media is the perfect way to get the message out, at the same time the show airs.

On Monday's episode, Katherine's character debriefs the President as the Prez reads her accounts from a stack of papers. Minutes later, the CIA tweeted a pic of the real Prez with the caption, "President Obama & other key national security policymakers now receive the #PDB [Presidential Daily Briefing] in a tablet form."

In another episode Heigl's character has a face-to-face meeting with CIA honchos. The Agency tweeted during the show that face-to-face meetings of this sort are extremely dangerous because enemies can target all of them at once. The CIA tweeted the preferable mode of communication is a "dead drop" ... where materials are left for the recipient at a prearranged location.