Are you certain it's the tube, and not a resistor, capacitor, or anything else? Check the power supply voltages with the mic connected. If any of the voltages are incorrect, shut the power off immediately and troubleshoot.
I just looked at a NOS NIB 5840 (RCA 9/60 vintage) out of the box and it has 8 leads.
Sounds like the unused leads were clipped in your mic. The 5840 isn't as expensive as a AC701, but with tubes getting rarer and more expensive, don't blame the tube first...You'd think twice about blaming the tube if you were dealing with a AC701 since replacing one of those is serious cash.