As the title says, I am looking for a real world example where a forward interest rate is negative.
Theoretically this is not a problem at all, if I look for a 3M forward interest rate that starts in 3 months from now I just solve for $r_F$ in the equation $$\operatorname{df}(Date1,Date1+3M,r_{3M})\cdot \operatorname{df}(Date1+3M,Date1+6M,r_{F}) = $$ $$\qquad \operatorname{df}(Date1,Date1+6M,r_{6M}) $$ where $r_{kM}$ is the $k$M-yield curve interest rate ($k=3,6$) and $\operatorname{df}$ is the discount factor.
It would also be interesting to see a reference to a negative yield curve interest rate.
A well known example of negative deposit rate is given on Wikipedia (Swedish Riksbank had an interest of -0,25% in July 2009).