Why is appending different streams to a file safe?

It's well known that redirecting standard output and error to the same file with cmd >out_err.txt 2>out_err.txt can lead to loss of data, as per the example below:

work:/tmp$ touch file.txt
work:/tmp$ ls another_file.txt
ls: cannot access 'another_file.txt': No such file or directory

The above is the setup code for the example. An empty file file.txt exists and another_file.txt is not a thing. In the code below, I naively redirect to out_err.txt both input and output os listing these files.

work:/tmp$ ls file.txt another_file.txt >out_err.txt 2>out_err.txt
work:/tmp$ cat out_err.txt 
file.txt
t access 'another_file.txt': No such file or directory

And we see that we lost a few characters in the error stream. However, using >> works in the sense that replicating the example would yield keep the whole output and the whole error. Why and how does cmd >>out_err.txt 2>>out_err.txt work?