I have tried both a 10k pull-down and pull-up resistor.

I was thinking a pull-down would be correct since the PIC gets a "true" signal, but as it didn't work, I tried the pull-up way; no success

Meanwhile, I have seen that the GPS module will always send the string my SERIN2 command is waiting for ("$GPVTG").

If the module has no signal (no satellite coverage), it will still send the NMEA sentence with empty data, just like this "$GPVTG,,T,0,M,,N,,K*xx".

So, if I disconnect the GPS from the PIC, it works but... it is not the way I would like to have it working.