This also reminds me that we audiophiles are using a particular definition when it comes to the word "natural" - one where we are mostly talking about matters of texture and tonal and harmonic balances, and here the word applies best to triode. If we expand on that definition a bit to consider other aspects that could also be described as being more so, then I would say that tetrode has the more 'natural' spatial presentation, dynamics and articulation, and ability to resolve fine detail. But we normally use other words to denote realistic performance in these areas.
I'll also mention an analogy I've used before to sum up the different impressions I get from comparing the two modes: With triode, I often feel as if the performance is less 'electronically' rendered than with tetrode, but also that I'm listening to it at some remove, sort of as if I were in the next room - while in tetrode, the performance sounds slightly more like an 'electronic' artifice, but also as if it's a performance that I'm more in the direct presence of, in the same room with if you will. Or put another way, with triode I more feel like I'm hearing the real thing but also that I can't 'touch' it, and in tetrode I feel more like it's embodied right there in front of me, but that it's less quite like the original thing and more of a slightly 'off' clone. The choice is like looking through a window at your very own mother (triode) vs. exchanging a hug with her replicant (tetrode). I guess people buy even better amps (presumably - hopefully, for me - at a higher cost) so the two qualities can be combined to an even larger degree than these amps can manage strong glimpses of in the respective modes...