Apple Watch Series 10 step count much lower than iPhone

If you are wearing your watch, it is the main source for counting steps, not the iPhone.

And, your watch arm has to move freely to record the steps correctly.

Some users did put the watch into their pocket, if moving the arm freely all the time was not an option.

Have a look at this discussion, for example:

My Apple Watch is not counting steps on w… - Apple Community

Your iPhone would move back and forth when it was in your pants pocket, causing the accelerometer to record steps.

Your Apple Watch is on your Arm. If that Arm is not swinging back and forth as you walk its accelerometer is not going to record steps. For example if you are carrying something, or moving someone in a wheelchair or gurney, etc... your Apple Watch is not going to be swinging and recording steps.

Chefs in busy restaurants miss steps and stand intervals because their Apple Watch hand is always carrying or holding something, even though they are on their feet moving about the kitchen all day long. Same concept as your work in the hospital.