Why would Craftworld Eldar fight Craftworld Eldar?

The Eldar are a dying race and they seek survival more than anything. It makes sense that they will avoid fighting their peers of other Craftworlds by any means. In particular, they will not hesitate to slaughter millions of "inferior" races (Humans,...) if it helps to save a few Eldar.

Does it mean there is no way Craftworlds could fight? I don't think so. As far as we know,

Each Craftworld is a self-sufficient, independent realm with its own culture.

(source: Lexicanum). As a consequence, we can imagine conflicts between different Craftworld. I couldn't find any example in the lore but here is a plausible scenario:

Remember that Farseers can only see some possible futures, and the better the Farseers, the more possibilities they can see, the further they can see, the better they can find ways to avoid a disastrous future. Also, Chaos influence can make future blurry and harder to read.

We can imagine a scenario where Craftworld A decides to attack a recruiting world of the Imperial Guard because their Farseers have seen that the IG will attack them a few centuries later. Problem, Craftworld B foresees that this attack will weaken the Imperium in that area and leads to a gargantuan Waaagh that will ultimately threaten numerous Eldar lives. Craftworld B needs to quickly stop their peers from Craftworld A before a catastrophe arises!

Let's look at some of the major Craftworlds to see potential clashes (all quotes from Lexicanum):

1. Saim-Hann:

   is regarded by other Craftworlds as anything from a slightly wild and dangerous place to a nest of barbarians living on the edge of Eldar society. They have developed the Path of the Eldar in a much more flexible way than other Eldar.

   Easy to imagine how other Craftworlds may consider some of their actions too risky.

2. Biel-Tann:

   is famous for its inhabitants' militancy (...) The Eldar of Biel-tan are honorable warriors who have taken it upon themselves to rebuild the lost glory of the ancient Eldar empire, through the destruction of the lesser races who have "usurped" the galaxy, and believe it is glorious to die fighting the enemies of the Craftworld.

   Biel-Tan's aggressive ideology would probably easily clash with the more subtle strategy of discreet interventions of other Craftworlds, especially Ulthwe during Eldrad's leadership.

3. Ulthwe:

   Ulthwé was caught in the gravitational pull of the Eye of Terror when it formed during the birth of Slaanesh and now orbits it (...). Being so near the Eye, the Craftworld has fought long and bitterly against Chaos forces. (...) The location of Ulthwé is the reason given by Ulthwé for maintaining such a large number of Warlocks, while others believe it more due to their location causing exaggerated psychic powers in the Ulthwé population that causes the emergence of more Warlocks. Whatever the reason, many Warlocks and other psychic warriors follow the Ulthwé armies to battle and their skills are even more advanced than other Craftworlds, being able to see further ahead in time and with greater precision.

   So, Ulthwe would be the perfect candidate for Craftworld B in my scenario above. Also, Eldrad had been seen to be closer (Inquisitor Czevac, warning Fulgrim of the incoming Horus Heresy,...) and interact more with humans than Eldar usually do. Most of it was out of interest but some other Craftworlds may be opposed to Eldrad's "liking" of other races.

4. Altansar:

   Altansar is an Eldar Craftworld that was caught in the Eye of Terror, the only escapee being Phoenix Lord Maugan Ra. (...) It has since returned but is now held in great suspicion by other Eldar, for how could any Eldar remain untouched by the predations of Chaos for so many millennia?

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In addition, the protagonist of the Eldar campaign in Dawn of War 2: retribution is the Alaitoc Craftworld led by Autarch Kayleth. During a mission,

The player character returns to Typhon Primaris to confront Kyras himself, only to be ambushed by Eldar from Craftworld Biel-Tan. Wary of a ritual they are undergoing, the player's character kills the Eldar there.

Note that the missions in DoW 2: Retribution are the same, whatever faction (Eldar, Space Marines, Chaos, Ork,...) the player chooses to play, so one may consider it as a necessary twist of the lore for the sake of the game and not a real example of Eldars vs Eldars battle. [Thanks for Connor Lonergan for the information].