After planet-fall, Uhura informs Captain Kirk that she is in contact with an Iotian named Bela Okmyx who describes himself as "Boss". Okmyx invites Kirk to come down to the planet's surface saying that a "reception committee" will be waiting for him upon arrival. Since the Horizon's visit was before the Federation's Prime Directive against non-interference, Kirk, Spock and McCoy are concerned about what effects the Horizon's crew may have had on the Iotian culture which was just beginning industrialization at the time and have a knack for imitation. The three beam down to find a culture resembling that of Chicago in the 1920s. They are immediately greeted by two men dressed as gangsters who threaten them with Tommy guns.
[from Memory Alpha]
In that intervening century since the Discovery left and the Enterprise arrived, Sigma Iotia II could have been visited by two other aliens... two aliens who normally couldn't be seen together.
Here is the picture that sparked this idea for fanfic:
"About a henpecked bank clerk who lives a dreary existence in suburban London. After a mix-up at a music hall, he finds himself onstage with ambiguously foreign magician Mendoza, who hypnotizes him into losing all of his inhibitions."
Jon Pertwee, on the left, and William Hartnell played Charlie Sterling and Detective Inspector Martin, respectively.
But that's in the Cineverse.
For the purposes of the Toobworld Dynamic, they're playing the Third and First Incarnations of the Doctor, respectively, Somehow their paths crossed each other's timestream and caused them to land on Sigma Iotia II. For their own safety, they have to disguise themselves as native Iotians and blend in, just as the First Doctor had to do when he found himself trapped in that Melkotian simulation of the Shootout at the OK Corral soon after the Enterprise crew had left. (But this time, they got to a 'Star Trek' location before the Starfleet officers.)
How the two Doctors ran into each other, what happened to them on Sigma Iotia, and which companions were traveling with each of them would be up to the fanficcer.
If you do take a stab at the story, send me a copy. I'd like to read it.