Finally! I've discovered what has caused all my pain and suffering the last days. It appears that there's some kind of bug in mikroC, in the Soft uart library. This means that I manually have to send a single 1 after transmitting a byte. Doing this, everything works like a charm! Woohoo! I really love mikroC, but it has a few bugs too many. Still, I always get answers from the developers and it has a lot of built in features I would never ever manage to code myself if I had to write in assembler.
I've tried various speeds. It seems that 38400bps works fine, but at 115200 some bytes are dropped. At 38400bps I'm able to transmit 4800bytes/sec, (a bit lower really, there's some overhead), or about 0.2 milliseconds per byte. It could mean a slight delay from pressing a key to the result is received by the second controller, but I guess I just have to try this out. I could also try to up the speed to, say 57600 and see what happens...
Update: 57600 works fine.
Update: 57600 does not work for receiving data when the receiver runs at 8MHz. I have to try this again when both uC's are on the real board.
Update: To use PORTA, it is necessary to add "ADCON1 = 0x07" to turn off analogue inputs.