> That's just incorrect though

Quite possible. But that's in no small measure because I have yet to find an actual cogent definition of "tensor" that distinguishes a tensor from an array. (I have a similar problem with monads.)

> So what I mean when I talk about the reality of the tensor I mean whatever it is the tensor is expressing in the physical universe

OK, but then "the reality of a tensor" not depending on the coordinate system has nothing to do with tensors, and becomes a vacuous observation. It is simply a fact that actual physical quantities don't depend on how you write them down, and hence don't change when you write them down in different ways.