How to choose the observation with the median value when there are an even number of observations?

Obviously, in most circumstances, you would just take the mean of the two middle values.

However, imagine a dataset of 100 countries. You want to know which country has the median household income. You could easily find the median household income by taking the mean of the two middle values, but how do you choose the median country within the sample?

Is there a rule for choosing one? Or is it appropriate to report both?

In Economics (to my knowledge) there is no such standard. In that case, I think it depends on: “Does it really matter”? Consider a sample of (say) 100 people, and you would like to find the median income, assume $n_{50} = 50.000$ and $n_ {51} = 70.000$ then a median of $60.000$ might seem “fair”, but really it is just interpolation. Then perhaps it makes sense to report both values (but I have not seen any paper doing that - at least in my field).
Perhaps $60.000$ is not even a realistic number - due to the specifics of your data. Suppose instead that the difference between the two is just $1$ - then I think most would agree that it does not matter (and you should be free to report what you would like).