[FurryToes]

I am getting this - only after putting new strings on.
I found this post while searching for a solution.
I put on D'Addario EJ45C Pro-Arte Composites - and use an Admira Artista classical guitar - bought new.
And I never had this problem until putting the new strings on - which suggests it's something to do with the strings. Tension/flexibility/internal structure?

I have to put my finger about half-way down the fret before the problem starts appearing. But I understand your description.
For me, there's a hint ("undertone") of 2 notes being played at the same time. A semitone below the note being played (which is obviously awful).
It seems to me that the string is too "flexible" - that holding it down, at a distance from the fret, is not pulling it over the fret strongly enough and so it's half-buzzing, half-harmonic-sounding.