I left some of the detail out of the original post because it didn't appear to be relevant to the point I was trying to make...but here it is anyway. A light 777 has a healthy rate of climb...as I backed off the climb (V/S) my colleague radioed the query to ATC. They told us to maintain altitude we were passing. I levelled the a/c at FL200, and that's when they said no known traffic. I could see on TCAS that the target was still descending onto us, but couldn't see the traffic. At this point, rather than wait for the inevitable RA, we turned off track...not an avoidance manoeuvre in the accepted sense, but a response to a hightly unusual situation. As we commenced the turn, we got the TA. We then saw the C130 descending...strictly speaking, perhaps we should havewaited for the RA..but I have to say he was coming straight at us and it seemed prudent to turn out of the way. What if he'd decided to switch off the tx at that point(i think they do it to stay stealthy, not sure)? Then he'd just have disappeared from our display and we'd have to visually avoid, not great with a closing speed of around 500kts...
Whatever. Maybe we did the wrong thing. That's not the point. The point is that it's unsatisfactory that these events are occuring under supposed radar control.