Thank you very much PoolDoc!

Here are my test results:
FC: 4.5ppm
PH: ~7
TA: 100ppm
CH: 800ppm
CYA: 100ppm
CC: 0

I was stupidly following my pool dealer advice and was raising CH for no reason. Now is very very high.
The testing at the second pool store indicated a high level of phosphate, just as you predicted. I do not remember the exact number.
The pool is now quite clear (a bit milky but definitely swimmable). No problem with algae yet.

I will appreciate any advice.