When the Romulan contact the Enterprise the second time, Kirk has it put through to the conference room. Kirk looks down to the three sided video screen and starts talking to the Romulan ship but there is no screen facing him. One of the three corners of the screen is facing him.
In the early second scene in the briefing room, where Spock reveals to the Enterprise officers that the Neutral Zone was breached on Kirk's orders, two officers are shown in the background. One is wearing a gold tunic indicating he is a member of Command or Navigation with commander insignia on the sleeve (holding the same rank as Dr. McCoy and Mr. Scott and outranking both Chekov and Sulu.) Another is a "red shirt" indicating he is a member of Security, Engineering, or Communications and that contains stripes indicating a Lt. Cmdr. ranking (the same as Spock) which would in turn indicate that he outranks both Lt. Uhura and Mr. Scott! No explanation or other identification is provided.
Two extras appear in the briefing room scene, one of whom wears a red shirt with a full commander's stripes. Who is this guy? Even if he's not an officer of the line (as McCoy is not), he would still outrank Scotty and would be the head of Ship's Services.
The Enterprise crew are shocked to learn that their surprise capture may be due to a new cloaking technology possessed by the Romulans. Kirk and crew already encountered cloaked Romulans two years earlier in Star Trek: Balance of Terror. However, the cloaking technology in Star Trek: The Enterprise Incident is new because it is improved. The Enterprise was still able to track a Romulan ship, though not accurately, in Star Trek: Balance of Terror and cannot detect them at all with this improved version.
Transporter beams cannot penetrate shields, yet Kirk is able to beam to the Romulan flagship and back. This means that the Enterprise and the Romulan warships have their shields down, an unlikely scenario with hostile warships facing one another.
After the Romulan Commander leaves Spock alone, he pulls out his communicator to contact Kirk. As a non-Romulan, it makes no sense that the Romulan Commander would allow him to keep his communicator - even if he is somewhat trusted by her.