>What I find interesting is that it seems the extra savings requires complex values. There must be something going on in the complex plane that is again over-counting with the naive approach.

"The rank of a tensor depends on the field over which the tensor is decomposed. It is known that some real tensors may admit a complex decomposition whose rank is strictly less than the rank of a real decomposition of the same tensor."

https://en.wikipedia.org/wiki/Tensor_rank_decomposition#Fiel...