Opening the Serial Monitor resets the Due, which resets the uP
using both the Reset inputs, and one of these two inputs resets
the RTC and other (unknown by me) stuff.
So, powering on, uploading a Sketch, or opening the Serial Monitor
all cause the RTC to be reset. If the Due will run in a situation where
it is always on, the RTC could be useful, if your Due board has the 32 kHz
crystal installed. Mine have it, but some pictures of the board show it missing.
To be really good support, you need to be able to read and write
the whole RTC-seconds value in one read (or one write).
I am not sure that this RTC library does that.
Has anyone looked?