I think
you would have problems with cross-talk between the
open tubes, unless you limited your design to a single tube,
or had large spaces between the tubes.
There are plenty of multi-section valves around, with several electrode
systems in the same envelope. Double triodes are really common, so are
multiple diodes + a triode (like the EABC80 or UABC80 triple diode +
triode (used in AM/FM radio sets for the AM detector, FM discriminator
and 1st audio amplifier). And triode pentodes (either signal pentodes or
output pentodes). In general the only coupling between the secions is due
to stray capacitance.
The glass does more than just hold the vacuum, it is also an
insulator.
Actually, it's not uncommon for it to be coated with a conduction film
for electrical screening purposes.
-tony