Priori Incantatem is a sufficiently "meta" effect that I think it makes sense as an automatic phenomenon when two brother wands fight. (Something like Incendio or Flipendo would be arbitrary, but spells that involve other spells or wands, like Prior Incantato or Expelliarmus, make sense.)

However, the only example we've seen of it is between Harry's and Voldemort's wands, which share feathers from the same phoenix. Look at the description of its effects from GoF:

I think that the particular light and sound depend on the wand's cores. If two wands with heartstrings from the same dragon fought, maybe they would cause a web of fire and an intimidating roar. If Fleur's grandmother gave two hairs to make two wands, maybe they'd attract the duelers to each other.

I'm not aware of any fanfics that feature Priori Incantatem with non-phoenix wands, but it would be interesting to explore the possibilities. Especially given all the non-canon cores people like to use. Has anyone done this?

 

From my recollection, Priori Incantatem is neither automatic nor is it a phenomenon.

It's a spell. Amos Diggory uses it on Harry's wand after finding it with Winky (and Crouch Jr) in GOF.

As for the excerpt, that's not Priori Incantatem. That's two brother wands refusing to fight against one another. That's probably more of an automatic phenomenon than Priori Incantatem is. Harry could have just broken the connection, but instinctively knew that it was not to his benefit. Instead, he decided to push the last beam at the end towards Voldemort's wand, and that's what caused Priori Incantatem (the shades of all the dead people.) Simply, Harry's wand had to overpower Voldemort's in order for Priori Incantatem to happen.

The sound and shape probably do depend on the cores, but to be honest if this happened between any wands but the holly and yew wands, I would probably just click out. It's not something that should happen or something that should matter. The only wands that should matter are the holly wand, the yew wand, and the Elder Wand.

 

The incantation Diggory used was "Prior Incantato". Dumbledore refers to the brother-wand phenomenon as "Priori Incantatem" (it's also the name of the chapter where it happens). Given Rowling's tendency to introduce some new magic early on and have it become more relevant later, I do think they're somehow related.

If you're saying that the light+sound effects and the prior-spell-ghost effect are two different things, then yeah, that's a good point.

Why don't you think it should happen with other wands? The Elder Wand is clearly special, since it can do things like repair other wands, but Harry's and Voldemort's wands are only significant because Voldemort made himself into a significant evil wizard (and then bound part of his soul to Harry by accident).

 

Priori Incantatem is the spell name where as Prior Incantato is the incantation. And, yes, I'm saying that the light+ sound is different from Priori Incantatem.

I don't think it should happen with other wands because wands aren't important unless they're important to the plot. Learning about Lucius' wand wood and core was important to the plot. Learning about Peter and Bella's wand was slightly significant in that it led to Harry questioning Draco's wand and the changing of allegiances.

Having another set of brother wands meet takes away from the connection/similarities between Voldemort and the BWL. While it is entirely possible for others to have brother wands, what are the odds of those two people a) meeting and b) dueling each other?

If some random OC were to start dueling Snape and their wands connected I would call bs and click out. I can't imagine where having this happen to anyone other than Voldemort and the Chosen One would be important.

 

No. "Priori Incantatem" is the name of the effect that occurs when brother wands meet:

This is similar to the spell with the incantation "Prior Incantato" in that both result in the wands producing images of their previous spells, but they are distinct things.

With regards to the brother wand effect, there's a couple of things we should take into account which modifies our understand of it from GoF:

- As of DH and Pottermore, we understand that wands are highly individual with varying levels of loyalty, different inclinations etc. As such it seems likely that different wands have different propensities towards spontaneous priori incantatem. Even Harry and Voldemort's wands, after all, did not always do this: they were able to exchange a number of spells without the effect occuring.

- As of Pottermore, we understand that Phoenix core wands are the most varied and temperamental:

As such it's entirely possible that the brother wand effect is restricted to wands with phoenix feather cores.

 

I like Suicune12's idea of a different Priori Incantantem effect for each core. It would only make sense for materials from the same animal, though I've no idea how you'd get two quality heart-strings from a single dragon. Looking at the wand core characteristics JKR released on Pottermore:

Unicorn hair - Consistent magic, bonds strongly to first user
Possible Effect - A consistent silvery bond & sphere, with a pure note rising to an intense pitch. Particularly protective of it's users.

Dragon heart-string - Powerful, easy to learn spells though can easily be committed to another owner
Possible Effect - A flickering, powerful dragon-fire red bond & sphere, with the sound of an endless roar loud enough to crush.

 

I don't think so. If you look back at the books, at least as much as I recollect, this brother wand effect happened when Harry and Voldemort's spells connected. I would rather say that a spell hitting a shield charm for example, doesn't count as connecting. It makes sense because having two spells head on collide with each other is somewhat hard, but the effect seems to at least have been studied, since I believe Dumbledore and Ollivandor both knew about the effect.

In short, I would rather think that this is a well documented effect when brother wands spells' connect, but because of the rarity of brother wands (No mention of such ever in the book besides HP and Voldemort, Ollivander seemed surprised and implicated that most animals only gave materials enough for one wand) and the rarity of spells actually connecting makes this effect rare. I wouldn't think it's phoenix feather cores only.