From "Death penalty opponents demonstrate in Schwarzenegger's home country":

Activists in California Gov. Arnold Schwarzenegger's home country voiced their opposition to capital punishment outside the U.S. Embassy in Vienna on Wednesday after the execution of convicted killer Donald Beardslee.

About a half-dozen protesters stood in the snow outside the embassy holding signs that read, "Schwarzenegger Terminates in Real Life," "Death PenaltyState Murder" and "No to the Death Penalty."

The Vienna chapter of Amnesty International urged Austrians to write letters of protest to Schwarzenegger's office, saying, "The death penalty is the most inhuman of all punishments..."

I hardly think six wacky protesters are going to change Arnold's mind about something. However, one can easily (depending, of course, on the individual) imagine cases where the interests of, say, Austria would be opposed to the interests of the U.S. If it were something popular and important, one likewise can imagine Arnold being called a traitor to his home country and perhaps legislation being passed to prevent him from traveling to Austria in the future. In that case it would create a conflict. Even if Arnold would make the right choice in such a case, there's no guarantee that others in the same situation would.