I am a relatively new manager in a highly matrix'd organization (multiple managers/accountabilities per person) and over the past six months my own functional team has been chronically suffering from over capacity work, out of hours nearly every day and all the related consequences.
My functional (direct) boss asked me to be more strict at prioritizing conflicting requests from other managers, but this would not help achieve the overall business objectives on time. We were clearly lacking resources.
That's when I identified people from other teams (including direct reports of my matrix bosses) who clearly had much more free time and were also quite interested in the tasks we didn't have enough resources for - and those are tasks that require very little training (just manpower)! I briefly spoke to the matrix bosses who agreed to get them working on it - also seeing an opportunity for them to have better control over their own requests.
My direct boss was quite happy about this and over time my team was able to focus on more important things (though we still sometimes work out of hours), while at the same time new talent was able to do focus on the other tasks without rushing as we did in the past.
For some reason, a few of the matrix managers recently started making remarkes such as:"Hey man, are you offloading this project to us, too? Can't you guys do it yourselves?"
I think this is a bit political, i.e. they are not conscious (or don't want to be conscious) of how much my team is overworked and instead only pointing out that we keep "offloading" our tasks to other people.
Another behavior I noticed is that some of these managers stare at us while we are having our lunch break in the canteen. I cannot make a perfect assumption, but what I can guess is that they think:"These guys say they are over capacity but take the lunch break allowance".
There is no rule anywhere saying that if we are over capacity we have to sacrifice our lunch break (cultural context: united kingdom).
How can I make it more clear to people outside my team that my team is genuinely busy, working hardest and - despite us getting smart and delegating stuff - we are doing it so that it can get done as best as possible?