In the right side up position, the exhaust valve on most regulators is at the bottom. The exhaust valve is small diaphragm exactly like a purge type snorkel. In order for it to work, it must unseat itself from the housing. When you breathe out, any water in the second stage case would sink to the lowest level and be forced out along with the air so you never notice it. In the inverted position, the exhaust valve is at the top, so when you breath out, a bit of water makes its way past the valve but has no way to get out. This is the water you are aspirating.
Years ago I worked in a dive shop that sold GSD regulators, they breathed wet in the upright position and dry in the upside down position….the exhaust valve was located at the top. Perhaps a GSD is what you need.