Tonga was much more steam laden / phreatic than either Pinatubo, Krakatoa, or Tambora; all three ejected far more tephra and sulfur.
The size and intensity of the Tonga blast came largely from catastrophic depressurization of supercritical water or highly pressurized steam, flashing over into an intense point explosion. It did vaporize the majority of the island and likely will reach mid to high end VEI5 all said and done, but it’s unlikely to have a major climate cooling impact as the other three mentioned did. They were substantially larger in terms of ejected volcanic material hitting the stratosphere.
With that said, a ~ -0.5C anomaly forcing in the Southern Hemisphere is still possible, but global effects are unlikely. Just wasn’t big enough in the sense of total material hitting the upper levels of the atmosphere as a huge portion of the blast was stream. No doubt the explosion was insane and the overall size of the mushroom cloud implied a much bigger eruption than what it will likely end up as.
I have seen the 400,000t sulfur measurement called into question, and am personally wondering how a portion of the eruption column reaching the mesosphere at over 55km may ultimately affect things, but in general a smallish negative forcing in the Southern Hemisphere should be about the worst potential impact of this.
Also should be noted the debate over the actual size of the eruption is still very much ongoing. A paper was just released characterizing the blast as a VEI6 which can have a much greater climate impact, but it depends on how much material hit the stratosphere (and beyond), some of which still seems to be the subject of debate.