No, we only have N-type electrons. It's notoriously difficult to P-dope vacuum.

I've always wanted to try that, make a reasonably high emission, low energy proton generator, and run that through a normal electrode structure. I expect Gm, perveance, transit angle, etc. are all approximately 2000 times worse than an electron tube, alone due to the difference in particle mass.
(And you thought GaAs PMOS was bad in comparison to its complement!)
In the sense that a device behaves like a tetrode or pentode, but is made of anything: we can cascode transistors just fine, and yes, doing it discrete is as good as doing it monolithic, indeed because we get into trouble if we try making the transistors too close together, which is known as an SCR. Making a bidirectional one would only require the addition of some diodes (to handle breakdown voltage) and connecting another anti-parallel.
Tim