Ken Pledger (Ken.Pledger@vuw.ac.nz) has offered insight that the standard spelling for English usage comes from the Latin, so rather than the Italian spelling "esplementary" we'll be using "explementary" for our software. He also pointed out the old word as an Oxford English Dictionary entry of "explement" meaning "that which fills up", but which had no mathematical use mentioned.
I have no hesitation to coin new terms, especially if there's a real use. Here it's for a CAD software toolbar button and function to switch between dimensioning the interior and exterior of an angle. The label will be "Explementary Angle".
In summary, we'll be referring to angle pairs as follows: Complementary angles complement each other to 90 degrees. Supplementary angles supplement each other to 180 degrees. Explementary angles explement each other to 360 degrees.
Everyone else can follow the usage, shout it down, or suggest a plausible alternative.