Go to a Renaissance Fair dressed up like the Doctor and look ridiculously confused about where you are and what’s going on. Mutter something about time being switched up, and how the technology you’re seeing doesn’t make sense for the period, and wonder aloud if you’ve made a mistake somewhere and merged two alternate timelines.
And if any of the staff catch on, experiment: is it possible that a Renaissance nerd, who, being a nerd, likely has a knowledge of other nerdy things, will be aware of what you’re attempting to do? If so, are they willing to take their Renaissance roleplay a step further and acknowledge your quirky roleplay, and not just pretend that they are a Renaissance-era wench, but a Renaissance-era wench who has met a time traveller and now has to cope with that fact? Could you, in fact, convince a random attendee or customer into being your companion for an hour purely by the force and bluster of your contrarian roleplay?
