I tried to understand my problem, and what I can see is that both GPS and Galileo are writing on the RX pin of the Intel Galileo Gen 1 board.
Here is an example of my problem, before connecting GPS TX pin to Galileo RX pin, I can read data on the GPS output (I'm using USB to serial converter), after connecting pins together (using switch), I can't read anything.
Is that a known bug ?
Yes, unfortunately the SoftwareSerial library is not supported by the Gen1. However, if you want to use several Tx/Rx interfaces an idea comes to my mind.
This is just an idea and I haven't tested it, but you can try using an analog mux similar to this one: http://www.ti.com/lit/ds/symlink/sn74lv4052a.pdf. So you would be able to manage more than 1 Tx/Rx pair with the same Serial interface that Galileo has.
Now, regarding your second post, how are you connecting the GPS device to your board? And why did you say both, GPS and Galileo, are writing on the RX pin?
Make sure the settings of the serial interface are correct: the baud rate, the stop bit, the parity bit, etc.