Sorry - wasn't sure where to put this -

If a DM was running a campaign (more or less) based on Forgotten Realms, and a player showed up with a Kalashtar PC, would you accept it?

Just to be clear - it wouldn't necessarily mean that everything related to Eberron would suddenly become canon in the campaign. Just the parts that mattered for a character to exist in that world.

Personally, I would—I do not think thematically they are all that alien too Forgotten Realms. At its core, a Kalashtar is a normal human (which the FR certainly has) with a soul taking refuge trapped inside.  You could either add some kind of dying world to the already complex Forgotten Realms cosmology, or just use the pretty darn awful place that is Shadowfell as the second soul’s homeland.

With very little effort you now have something which is effectively the same as a Kalashtar and could work with their mechanics, but firmly rooted in FR lore. You do not even need to make them an inherent part of the world—pretty easy with Kalashtar to say they are a unique combination of souls.

I'd allow it, suggesting, that the character is connected to a spirit of the outer planes of mount celestia, bytopia, elysium or arcadia. As the description says (virtuous) "spirit of light". (Or simply adding the plane of dreams as a demiplane.)

You can also just say that their family found their way from Eberron to Toril. The character might even have a trinket from them - maybe later you can grant them access to the Dream of the Blue Veil spell and they can use the trinket to visit Eberron for a short while. Or maybe take up spelljamming and the character might want to visit home (spelljamming and the spell both allow you to transport to other material plane worlds like those found in forgotten realms, eberron, greyhawk, theros, etc - can make for some great adventures).

Given all of the weirdness of the FRs any transplant is fairly easy. If nothing else the character can start by finding themselves transplanted suddenly from Ebberon to Waterdeep after walking through a portal one foggy morning (actually Halaster has foreseen them doing something significant or is just playing games taking folks from various worlds to see what they do). That or some ancient portal misfired as it died and opps you switched worlds. The FRs are great that way - with 35k years of magical history and 25k years of elven history all with high/epic level mages making portals and gates and stuff you can pretty well bring anything and anyone you want into an FR campaign if the DM is willing.

There's also the old standby of "an enclave of this race is found hidden in an uncharted region."  I recommend an island or archipelago located somewhere between the Sword Coast and Mazteca, maybe the survivors of a Netherese experiment escaped there and hid themselves from the rest of the world.  Then you just have to explain how your character ended up meeting the party.

Toril has many portals.  It’s not unreasonable to believe that a few are to Eberron.

Since you say it's more or less based on Forgotten realms, there shouldn't be a problem. He could also choose the Kalashtar as a mechanical corset, but be something else.
Now, officially there is no Kalashtar in FR. Could one have traveled from Eberron somehow? Well I guess so. Somehow. But Eberron in particular is quite isolated. Perhaps it would be interesting to consider how and why that specific individual has traveled from Eberron to Toril, and see if it's the fantasy that you want to play. If all he wants is the mechanical benefits of his race, he can say that he is a human with those abilities for whatever reason. Or anything else you can think of.

Maybe just me, but I'd reflavor it to be just a personal condition. This one person has a second soul inside them. Or maybe they're just psionic. Or maybe they're the child of a night hag. I see this kind of thing as a great opportunity to come up with a cool character backstory. The DM might even find a way to tie it into the campaign.

People cling to official lore way too much. In game terms, Kalashtar is just a handful of mechanical benefits. Any way you can find to describe those benefits in the light of your setting can work, and often your setting is better off with that tailored description.

I have one in my campaign right now. The player is a kid so didn't put that much thought in to it when choosing, them and have a classic "don't know where they are from or who their people are orphan" backstory, so my "in world" explanation is that he is a member of a small group of kalashtar that fled Eberron to Toril at an indeterminate point in the past and have established a hidden colony somewhere in the Sword Coast. At some point im going to set up a quest that can lead him down the path to find and connect with the colony but haven't yet, they are in the middle of several plot lines already.

Not only would I, but I played one while going thru Rhime of the Frostmaiden. I love the idea of Kalashtar, and worked with the DM to come up with a backstory that fit his campaign (and during the campaign he sprinkled in run-ins with the evil Quori.

I played the female Kalashtar (Twilight Cleric) as secretive, on the run, and never talking (only relying on her telepathy with those she trusted, and pretending to be mute to those she didn't). Was a wonderful character. I played her focused on defensive abilities and buffing. It was challenging because I almost always end up being the 'face' of whatever group i'm in, playing a submissive, diminuative, character in hiding and on the run who never speaks was a lot of fun.

And if you're having trouble marrying Eberron with Toril, one thing I frequently do, is simply say that the other campaign settings are just different continents on the same world - and people have travelled between them and settled.

You might say that kalashtar descended from those who lived in the Jhaamdath psiocracy.