Getting back to this question:
I dont know for certain, as I have not played with readers much. But I can speak about serial coms in general:
*MOST* of the time, two devices (plus the PC) on one RS/TTL232 bus does not really cause much issue. Especially when they are close (such as using connecting a XBEE or Bluetooth module to an Arduino, sharing the 232 bus with the FTDI USB chip to the PC).
*BUT* 232 is specifically NOT specified for more than two devices (endpoints) for two major reasons:
1: The UART has no specific hardware (and software is generally not implemented) to discriminate messages for specific devices. So, your reader may be hearing stuff It doesnt understand, and simply decides to stay out of it. Think of it like a network collision. "Im not in with the in crowd!"
2: On longer runs, you will get all sorts of nasty signal reflections and messy bits when you try chaining multiple serial232 devices. Most multi-drop networks have termination resistors at each end of the chain to clamp down the signals so that they dont bounce back. Serial232 specifically does NOT have these, because it is never meant to have more than two devices anyway. So, according to the hardware, the signal goes in and never comes back out. Depending on the hardware design you have, one or more of the UARTS on your chain may be 'sucking the life out of' the signal (because that is exactly what it is specified and designed to do. Remember that voltage levels on a 232 net are not set in stone. For TTL high signal could be 1.5, 3.3. or 5 volts. For RS232 it could swing between -3 and +15 volts, but 0 to 5 volt swings are common on crappy USB-serial adapters).
Unfortunately, its one of those things where we get away with it a lot of the time. Especially with embedded hardware, and notoriously so on Arduino/radio chip combinations. But it is technically against spec and highly unadvised.