I think I would wait for the other person to 'volunteer' the information. This is because I believe that mixed race people can get tired of people asking them about who they are 'really'. It certainly depends on the person, though.
My children are mixed race and they would probably not mind being asked, for example. Still though, I think that if someone is getting to know you, they will tell you about their roots when they feel like it.
Any other opinions on this?
Ps - we can't say - what's your other half race.
I would just ask ' where are you from'?
This gives him the chance to say 'I'm from Japan' or to volunteer more information. It gives him the choice of how much he wants to share.
My kids get asked this question all the time - because people see an Irish woman with Greek kids and they get confused or curious.
My kids say we're Greek-Irish.
Just like saying African-American etc.