Ventilation may not cure this ill.
First, measure the humidity inside the coop and outside the coop.
If it's within 10%, not much you can do.
The big issue is, cold tin/tarp and warmerish air = condensation.
I have a standard roof of 1x boards under asphalt shingles, and tons of ventilation, still get condensation on underside of roof if there's snow on roof and thaw type temps.
I don't know why you put a tarp under the tin, but, bad decision.
Foam insulation under tin is the way to go.
Not sure of a quick fix either...might tack up some foam boards on underside of rafters,
but can't really see that part....and cant' see where you roosts are.