How can I find those true mudguards that rental bikes usually have? I don't know what terms to search for or which shops to look in.
These images show the fenders I am looking for:
Bicycles Stack Exchange is a question and answer site for people who build and repair bicycles, people who train cycling, or commute on bicycles. It only takes a minute to sign up.Sign up to join this community
The side guards on the rear wheel are called "skirt guard" or "dress guard". They are commonly found on Dutch bikes. Apparently searching for the Dutch term "Jasbeschermer" is also helpful and gives you products like this:
They are mostly intended to stop loose clothing from getting tangled in the wheel. This only makes sense if you also use a chain guard.
If you only want to stop the spray by dirt/water, but don't need the tangling protection, use regular full-length fenders: