Can Humans transformed into Pirmals temper?

[Lunair]

Essentially, what happens is in cases where a person is exposed to aether to the point that their body cannot physically hold more, they undergo corporeal corruption, with the way this manifests differing depending on the aspectation and polarity of the aether in question.
In Oversaturation, because no one aspect is dominant, crystallization occurs, which we see with the Crystal Exarch, as well as numerous examples of plantlife in places such as Mor Dhona or the Isle of Haam during Aetherfont.
In Overaspectation, the corruption often results in mutations and transformations that are thematically related to the aspect in question, resulting in beings such as the Drowned. Notably, this is also the reason that Black Mages tend to die without the soul crystal, since they become overaspected drawing in fire or ice, and corrupt in a related way. This particular flavor however seems to be able to be influenced by a Primal to some extent, further twisting the resulting corruption as seen in the case of corrupted telophoroi.
Overpolarization however is a result of overexposure to a single polarity to an extreme. However, being a measure of energy, Polarity equalizes much faster than aspect does, so you typically won't see corporeal corruption from Overpolarization outside of environments that're already steeped in the polarity, such as the First or Thirteenth, (or theoretically the deepest reaches of the Lost City of Amdapor, but that's speculative.)but in cases where it occurs, you get what're commonly known as Voidsent or Sin Eaters, depending on the polarity.

[Lunair]

It's worth noting that all three of these conditions will trigger aethersickness before they reach the point of corporeal corruption, as the body becomes more and more suffused with aether and get nearer and nearer to that limit.

Additionally, it's also worth noting that, as mentioned repeatedly by G'raha Tia particularly following Shadowbringers during the efforts to cure tempering, you cannot un-corrupt someone who has become corrupted corporeally, (theoretically, an ancient would have the capacity, but it'd take that capacity for creation magics to be able to "unmake" the corruption.) but it's worth noting that Primal Manifestation is not a form of Corporeal Corruption.

(While i'm sure someone's going to point out Zero stops being corrupted looking during the post EW questline, she's a hybrid, which is a WHOLE can of worms that would require an entire other post to get into but it ultimately boils down to "she cannot be used as evidence for any other cases that aren't also hybrids.")

Regarding the actual causes of Tempering beyond the aetherial mechanics behind it, Tempering by most modern primals is a result of Ascians tampering with the ritual involved, as revealed by the loporrits during the effort to power the Ragnarok with the Mothercrystal. Primals summoned without the ritual, accordingly, would lack the compulsion to temper, though they'd still have the capacity to do so, and it's highly likely that extremely powerful primals would temper people unwittingly purely due to the sheer amount of aether exposure unless they made a deliberate effort not to do so, which is likely what happened with Zodiark before they were sealed.

As a result of this, it's logical to assume that anyone who summoned a Primal without having the Blessing of Light (also known as the Travelers Ward that Hydaelyn gives to anyone who hears her voice, (which seemed to be somewhat automatic anytime someone awoke the echo) would become tempered due to quite literally being bathed in the aether of the Primal in question. Depending on the Primal, however, the outcome of this tempering would differ.

In the case of the Archbishop, being the head of state and focal point of the Ishgardian Orthodox Church, it's highly possible that they were tempered simply by taking on the office of archbishop, (due to being the focus of the collective faith of Ishgard) though the actual mechanics of this are ALSO an entire other post's worth of explanation relating to lore we got from the Myths of the Realm raid series and other information we get about the way collective faith influences Primals. They then went on to "lightly temper" the Heaven's Ward to obey them.

[Lunair]

In the case of Tsukiyomi, while Yotsuyu was almost definately tempered by the transformation, i'm not sure that they'd have gone on to temper anyone else, though she definately would have rampaged in Doma given the opportunity. This is largely because of how the collective faith of the Far East was concentrated on the mirror that served as the catalyst for her transformation, the same phenomenon that contributed to Susanoo manifesting of his own accord once sufficient faith (concentrated in the three Relics that were involved in him manifesting) was present in the same location.

Y'sayle meanwhile likely wouldn't have tempered anyone. Due to the Echo, she should theoretically have been fully immune to the influence of the collective faith that shaped Shiva, but it's worth noting that her followers may well have been tempered anyway just due to proximity depending on how often she manifested the primal in their vicinities.

In a related note, Vauthry's control over Sin Eaters would have essentially been a form of tempering being used as pseudotelepathy, at least in theory. As far as differences between Sin Eaters and Voidsent go though, i'd largely chalk that up to the nature of the polarities, and the time between their formation and the WoL arriving in their respective shards. But those differences make it hard to say for certain if Sin Eaters would have developed more coherent speech over time. Personally, I suspect it's purely a matter of the properties of the polarities, since the two Light wardens we encountered with actual sapience were both spoken who presumably would have had sufficient control and power to overcome the static nature of light.