Possibly the most famous example of fictional physics in anime would be the Minovsky particles and Megaparticles from the Universal Century Gundam timeline. Megaparticles are easier to explain; so called because they are large for a subatomic particle, their properties are the explanation for most of the more futuristic weaponry.
Minovsky particles are more interesting because they show how committed Tomino was to justifying what he thought needed to be justified, even though he could have easily gotten by with no explanation for this. (But Tomino’s ideas of what was and wasn’t realistic weren’t always right. The Guntank, actually more realistic than walking mobile suits like the Gundam, was phased out for being “too super robot” just because Getter-3 also had treads instead of legs.)
You see, during the Cold War, everyone was far more concerned about long-range missile strikes than about any possibility of armies fighting each other, and there were predictions that future wars would consist entirely of missiles being fired from far beyond visual range. Well, the idea that infantry and tanks would become obsolete haven’t come to pass and quite possibly won’t, but it is true that naval and air combat both typically occur at ranges measured in miles, and even on the ground snipers are more important than they used to be. So Gundam had the Minovsky particle, which interfered with long-range sensors to such a degree that all battles had to take place at closer range.
But think about it. The main purpose of Minovsky particles is to justify something that essentially all sci-fi battles in visual media do without being commonly questioned. Babylon 5 might still be the closest thing to hard science fiction in television history, and certainly the Starfury was the most realistic space fighter in fiction, but even it had space combat at close range.
In any case, the Standard Model of physics doesn’t really allow for undiscovered fundamental particles, but it turns out Gundam didn’t really need to invent one for this purpose anyway. The primary effect of releasing Minovsky particles is the same as an electromagnetic pulse. (Electronic systems on ships and mobile suits have to be shielded so they aren’t disrupted.) Of course, Minovsky particles are also used in beam sabers and I-fields, which are a topic for another post.