The pictures will help but my first thought is matching stone as close as you can find. Pulling out random existing tile along that edge so that new tile meshe's wth old will help.

@klem1 I will try to update some pictures today, my flat is occupied after a 5 years rental agreement, I meet with the tenant today.

@Sophie Wheeler: I asked the administration, and since they are fully recessed balconies, moving the windows further out should not affect the appearance of the entire building. If it was a cantilever or another sort, yes, I agree this could be an issue. The window also covers the entire opening, so it should be pretty straightforward. 3 ft will also go unnoticed from the perspective of someone outside.

If wood be acceptable - there are wood-looking tiles. It's hard to tell the difference between them and real wood.

I you need a transition strip - check out Schluter.

Schluter

Cheers guys, I found some pictures of the combination of stone tiles and wood. In Chile we are very keen on using elements of the colonial architecture with new materials. In the pictures you can see stone tiles with "demolition wood" (wood from demolished old colonial houses). I think I can do it without a transition stripe, I have a table saw and router to make everything to the millimeter.

I will check again, maybe I have one or two tiles at my cellar. If I do, I could do something like in the picture.

I finally took some pictures of the living room. As you can see, the entire opening is covered by the window, and it is a recessed balcony. I saw from the street, and the windows are not visible. I read the contract, and it only prohibits changing the colors of the balcony's walls, putting satellite dishes and closing the balcony, the administration said that moving the window further outside should not be an issue.

I also found three complete from the original remodelation tiles in the cellar, That means I could make something like in the pictures above, combining wood and the same tiles from the rest of the room.

I know, the tiles on the balcony are awful, I will replace them later though.

What you lack in conversational English you certainly make up with photography skill. Those two shots say it all and I'm sure desighners will be weighing in with solutions after seeing those pictures. Meanwhile I'd advise, no te salga yur trabajo del día