Of course in real life we are often interested in small patches of spacetime in which the state approximates the vacuum in the above sense. E.g., what you get when you take a jar and remove all the air. However, regardless of what we call them, they're interesting states. All the basic notions of thermodynamics apply. If the cosmological constant is nonzero, then the vacuum has nonzero energy density and pressure.
Negative pressure is the same as tension.
Usually this is assumed to be isotropic. However, if we start worrying about how this nonzero energy density and pressure affect the curvature of spacetime, we can easily find ourselves trying to understand quantum field theory on curved spacetime, which is one step up the ladder of confusingness from plain old quantum field theory - and which suffers from the lack of any experimental data to test our theories. And if we start worrying about quantum fluctuations in the curvature of spacetime, then we're doing quantum gravity, which is yet another step up the confusingness hierarchy - and again suffers from the lack of experimental data. So I'd suggest that you ignore this paragraph and stick with plain old quantum field theory, which is already plenty interesting. Mixing these up leads to endless confusion.