I've also seen an interview video made while both groups were on tour in the UK. The YouTube uploader claims it's from 1987 which tells us that the tour went to more countries than just the US. However, the video could be from Run DMC's Raising Hell Tour which the Beastie Boys supported them on in 1986.
The video is here:
The video only shows USA dates so I'm not sure if they toured other countries together or not.
Here's another interview with the two groups in which the tour is specifically mentioned:
If anyone has any extra information on the tour then please let me know in the comments and hopefully we can build a decent Wikipedia article on this seemingly missing part of both groups' history.