Unable to catch/export some cookies

Question

In attempt to automation of some job on a certain site, I use php-webdriver. Target website is written in Vue. For certain POST query this site uses cookies, but this cookies does not present nor in Web Developer Tool > Storage > Cookie, nor in dump via $driver->manage()->getCookies(). I can see this cookies only in certain POST queries. Also, I have used Tampermonkey for logging (set|get)cookies events but w/o luck... I was thinking of using Tampermonkey for catching and saving this cookies.

I have not seen this "hidden" cookies ever.

So, need advise how to catch this cookies.