I've read that the spheres behind the tower on the bridge are for satellite navigation. Is that correct? If so, why are they so big when commercial satellite navigation receivers are small enough to fit into a phone?
Most civilian mobile phone GPS are not true GPS. They rely mostly on the mobile relay towers on the ground near them for their GPS functionality (why GPS typically doesn't work when phones have no mobile signal).
Any true satellite mobile phone is instantly recognisable from conventional mobile phones from the massive additional receiver they need to pick up satellite signals directly.
As a general rule of thumb, the bigger the receiver, the weaker a signal it can pick up.
True satellite phones can get very choppy and cut out if there is bad weather.
Storms can get far more server out in the open oceans compared to on land, and it's during such times that you will need you satellite navigation to work flawlessly the most.
Something else to factor in is the fact that unlike land vehicles, naval ships can pitch quite heavily during rough seas. As such, it is beneficial to have a much bigger antenna such that even when the ship is pitching heavily, enough of the antenna still has direct line of sight to enough of the sky to have a good chance of picking up GPS signals to avoid loosing contact.
That is why naval GPS antennas typically are beefier than for land based versions.