Has anybody ever had such a problem???
Could it be coming from the mast????
Yes, and yes.
When I started chasing down where water was coming from, I isolated the bilge-pocket just aft of the mast from the main bilge (small wooden bung in the limber hole did the trick). I was somewhat astonished to learn that more than a gallon of water comes down through the mast into the boat in an average rainstorm (hey, there's probably a reason they call this place the Pacific North Wet). But... at least I knew how much of the water was from the mast.
The rest of the water.... is still a partial mystery. When the boat is in the slip, the main bilge gets some water in it. I believe it, too, is rainwater. On a whim I taped over the edges of the starboard lazarette lid, and the outboard scuppers, and after the next rainstorm the amount of water in the main bilge was cut by about 2/3rds. I suspect there is a crack in (at least) one of my scupper hoses, and I also suspect that some weatherstrip or similar around the edge of the lazarette hatch may make a difference. So chasing those down is on my project list. I've never seen any trace of dripping or weeping around any of the thru-hulls, I don't see water dripping from the dripless gland, and I don't think I have a keelbolt issue (the bilge stays dry when there is no rain), so... yeah, apparently rainwater has a variety of paths into my boat.
It's a journey, not a destination.
Underway... I think there are other things. I don't suspect my dripless, and I don't think it is my rudder gland. Even if the bilge is dry when I leave the dock I often end up with water in the bilge when under way.
As an example, I motored for a solid 5 hours last August on the way across the Sound, and when I got where I was going I found my bilge almost full. Which... bothers me.
I have a strong suspicion that, as you found, when the boat is either heeling or at hull-speed, the "exits" under the stern have the potential to become "entrances". I haven't figured out how to test it yet but the electric bilge-pump hose is my prime suspect, both because it is the thru-hull closest to the static waterline, and because there is no check-valve or anti-siphon loop in the path. My project list for the spring includes replacing the hoses to all 4 scuppers, and putting (at least) a loop in the bilge-pump hose; more likely, if I have the time, I'm going to replace the existing system with a diaphragm pump (which acts as its own check-valve).
It is always clear water, except one time it was black.
I had that, too. First sail in the boat after it had been idle for several months while I worked on the mast. Water in the bilge was black and stinky. My guess is/was that it was water trapped under sections of the TAFG, which only found a path out to the main bilge when the boat heeled. I don't even want to think about what was growing under there ("creature from the black lagoon"?). I ended up funneling a bunch of water+bilge-cleaner into the hole in front of the mast, and pumping it out when it found its way to the main bilge. Cycled this solutions through a number of times, in an attempt to flush out any grunge under the grid. Have no idea whether it really did anything, but at least now my bilge-water is clear and smells like bilge-cleaner...
Bruce