Dear:
I think that implement handshaking will be hard for you, if you have not experience with it.

But taking a quick look at your 8051 soft is very very easy to see that:
YOU NEED A DELAY BETWEEN DATA SENT !!!!!!!!!!!!!!!!!

try this please:

DATA_OUT: 
lcall delay
MOV SBUF,R1 
JNB TI,$ 
CLR TI 
CLR TR1 
SETB TR1 
lcall delay
MOV SBUF,R2 
JNB TI,$ 
CLR TI 
CLR TR1 
SETB TR1 
lcall delay
MOV SBUF,R3 
JNB TI,$ 
CLR TI 
CLR TR1 
SETB TR1 
lcall delay
MOV SBUF,R5 
JNB TI,$ 
CLR TI 
CLR TR1 
SETB TR1 
lcall delay
MOV SBUF,R6 
JNB TI,$ 
CLR TI 
CLR TR1 
SETB TR1 
lcall delay
lcall delay
JMP DATA_OUT 
END 

Then make the routine delay with some milliseconds and VOILA!!.
After you get it works fine ...play with the delays to optimize spped.
Good luck.