...Except... Shortly after the Doctor and Vicki leave Ian and Barbara are captured by slavers and sold into slavery. Thus ensues a dual story line, the two couples embarked on disjoint but nearly crossing journeys. The Doctor and Vicki are on their way into Rome and fall into the murderous intrigue of a plot to kill the Emporer Nero, while Barbara becomes a "handmaiden" to the Empress of Rome, Poppea, wife of the Emporer Nero, and Ian becomes a galley slave and later escapes and goes to Rome in search of Barbara.
Ah.. the splendor and intrigue of Ancient Rome. We see it all.. poisonings, a vengeful Empress, a lustful Emporer, scheming sycophants, galley slaves, slave revolts, the Rome Coliseum, ..etc..
Coincidentally the TARDIS crew lands just before the burning of Rome, which the Doctor unwittingly facilitates.
All through the story the Doctor and Vicki and Ian and Barbara come close to crossing paths and come close to meeting each other, but they don't. In the end the Doctor and Vicki return to the Villa to see Ian and Barbara lounging, recovering from their own stresses and adventures, but believing that Ian and Barbara never left the Villa.