Drainage Problem from Bathroom Sink

I have a bathroom with a sink, toilet, and shower all in a row next to the same bathroom wall. Recently, water started leaking out of the wall between the shower and the toilet. Also, it started leaking into a bedroom that shared that wall.

After much trial and error, it turns out the leak is caused by running water in the sink. The sink eventually starts filling up (partial blockage?) and the water starts leaking out from the wall. The sink drain pipe goes into the wall. The piping layout is that the water eventually connects to the big drain (from all three) which is outside the house (beyond the shower).

When the leaking happens, the shower drain does not fill up but looks fine.

So, it looks like my sink has a partial block...fair enough. But why is water leaking out of the wall? Is there a drain vent somewhere (inside the wall and lower than the sink)? This could get to be a fairly big problem quickly because the house is 1950's with a tile up to six feet high in the shower that links directly to tile on all walls of the bathroom up to 4 feet high. The only saving grace might be that the bedroom wall is not tile.

Anyways....any advice or help would be appreciated.

Its probably leaking at a cast hub inside the wall or the pipe itself could be rotten, you will need to access the pipe and have it repaired.

The way a plumber would do this is open the bedroom wall from about 24" high from stud to
stud down to baseboard and you will have a perfect view as to what pipe is leaking

then post the pictures and we can tell you what you need to do to correct problem !

Since the bedroom wall already has water damage no loss and you don't mess with
the tile walls !

The pipe from the sink probably goes sideways, and it has deteriorated. It could be a copper pipe which has eroded from drain chemical usage, or a plastic one with a broken fitting. It could be other materials but they do not usually fail.