Tour of Duty is a Bronze trophy rewarded for completing one Co-op mission in every region in Resistance 2. There are 7 regions: Chicago, Orick, Axbridge, Bracknell, Bryce Canyon and Holar Tower. You will unlock this trophy as a natural progression towards the Specter Intel trophy.
You need to complete 5 Chicago missions to unlock the Orick region. You need to complete 5 Orick missions to unlock Axbridge. Three Axbridge missions unlock Bracknell and 3 Bracknell completes unlock Bryce Canyon. Finally you need to complete 10 Bryce Canyon missions to unlock Holar Tower. This is a total of 26 mission completes before you unlock the last region.
At this point, you only require to complete a Holar Tower mission and you will have completed one mission in each region.