Flirty Doctor Who TV ads from the 1980s suggested the Fourth Doctor and his companion Romana were an item. As a rule, during the classic Doctor Who era the BBC frowned on the idea of hanky-panky in the TARDIS. There were good reasons for their concern; by the 1970s and 1980s the show's format had changed, and as a rule the Doctor no longer traveled with groups. Instead, he was usually accompanied by a single attractive young woman. The BBC watched carefully, wary lest there be any hint of impropriety.
And then, in 1979, Lalla Ward joined the TARDIS as the second incarnation of Romana. Viewers immediately noticed the chemistry between Tom Baker's Fourth Doctor and this new Romana, and it made sense to consider them an item. After all, this relationship was unique in the history of Doctor Who: Romana was the Doctor's equal, an actual Time Lady who was just as intelligent as the Doctor - possibly more so. Although the show didn't address it directly, this was a sea-change in Doctor Who lore, because for the first time viewers were speculating there could indeed be more than just time travel going on in the TARDIS.
And then, in 1979, the BBC licensed Doctor Who-inspired adverts for Prime Computers in Australia. These were never broadcast in the UK, although they were attached as special features on the DVD release of "Destiny of the Daleks," and consisted of four ads featuring Tom Baker and Lalla Ward. Absent BBC oversight, these ads did indeed suggest a romantic relationship between the Doctor and Romana.
The adverts are quite dated - as are the Prime Computers they're encouraging viewers to buy - but they're most notable for the dynamic between Baker and Ward. The Doctor and Romana are shown in far closer physical proximity to one another than in Doctor Who, and in one Romana snuggles her arms around the Doctor as he chatters about ideas for using the Prime. "Ask it how to handle a woman," she tells him, visibly frustrated he isn't responding to her affection. The Doctor does so, and the Prime gives a simple response: MARRY THE GIRL DOCTOR. He proposes on the spot, and Romana agrees.
Viewers weren't deceiving themselves when they noticed the chemistry between the Doctor and Romana. In the real world, Tom Baker and Lalla Ward married in 1980. Unfortunately things didn't work out, and the marriage only lasted two years. And while the Doctor has certainly had romantic relationships in Doctor Who since, Romana and the Fourth Doctor were never officially an item in canon.