"my watching" is a correct and usable phrase in written English.
It can be used to refer to something that you have been observing or monitoring over a period of time. For example: "I have noticed an improvement in my health since I began my watching of what I eat every day."
Exact(10)
I thought that my watching partner would share in my frustration and approve my motion for skipping.
"Ah breasts," said one of my watching companions, "it's been a long time since we saw those in Game of Thrones".
When the ball came loose out there, second base umpire Dana DeMuth signalled that Kozma had held it long enough for the force, even though everyone in the northern hemisphere, including my watching fox terrier and I, could plainly see that Kozma had barely touched the toss with the tip of his glove.
They were so entwined that I was unable to separate the atmosphere of one from the other, to not let book-Jack's motivations bleed into my watching the movie, to not let the horrifying twins appear as I read about the hotel corridors.
Over time, our watching it ironically turned into my watching it in earnest, alone.
You'll be glad to know that it takes less than 30 minutes of my watching people and rotating the same box of Whiskers to rouse suspicions.
Similar(48)
"I had to keep my eyes down, watching my step.
With my mum watching.
Item 34 Since receiving HEN, my family watches over my food.
Skilling availed himself of the "not-on-my-watch" defense.
"My watch stopped!
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com