Another device use RS-232 port this half-duplex port mode
Have you thought that all you should need do is code up your comms routine so that you don't send anything until you get a finished response back for the device.
Doing this will give you half duplex.
Remember you have control when you send data out the serial port.
I am assuming the device you are communicating to can only do half duplex, is this correct?
Also, I am assuming the device you are communicating with will only respond on Arduino commands and not at any random time, is this correct?
You need to be very clear and specific with your descriptions on how your equipment works and how you wish it to work, or everyone makes lots of assumptions.