Not either of those, I'm afraid.

More about the virtual reality:

The places you visit in a VR setting are historic sites that you also visit in the present time in the game. The VR simulates their past appearance (and I'm talking centuries past).

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I seem to remember playing a game like this 10 years ago?

It is called something like Templar Knights or Knight's Templar; I think? Anyway, that's my guess..

You're getting warmer, Karsten! It is from around 10 years ago.

Hopefully this next one will give it away to whoever's played it:

Set in Turkey

Is it Byzantine: The Betrayal, Jess?

Journeyman Project 3?

LadyKestrel, you got it.

The floor is yours.

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Bump, bump, bump

Sorry for the delay, everyone!

First clue:
My name is well suited for an adventure game.

Kate Walker in the Syberia games?

That's a good guess, Bernard, but not the character I have in mind.

Next clue:

I'm not handsome, but I am clever and kind.

Gus MacPherson - Post Mortem?

No, not that one, Zookeeper.

Next clue:
I have traveled quite a bit so far. When I'm away, I write long letters to my wife, whom I love dearly.

Agon - but not sure which one, so I'll guess Agon 4.

That's good enough, Tina, since it's a series. Professor Hunt is one of my favorite characters. Part of his appeal is the excellent voice acting, at least in the first 3 episodes. (I haven't played #4.)

Okay, it's your turn.

Thanks! I'll be right back!

Cookies and water again?

Sounds like a Nancy Drew.

Sounds like Next Life

Over to you Shany.

Simon 4???? (Is there an English version yet?)

Not that I know of, but that's not it anyway.

However:
This game has something in common with Simon 4, and with Nancy Drew: Treasure in the Royal Tower.

East Side Story?

You got it, Becky.

Thanks Shany!

The clue below refers to one word in this game's title.

Clue 1: baleful.