I have a little collection called 'the Waltz Book' by Bill Matthiesen, and on page 46 is a tune called 'Spanish Waltz.' The B part of the waltz is the music for the chorus of 'Green Hills of Tyrol.'

The A part of the song and of the dance seem unrelated.

Matthiesen says "The tune goes back at least to the 1820's but is still popular at vintage dances today to accompany a waltz-circle mixer of the same name."