RF switches for RFID HF 13.56Mhz
simulate this circuit &ndash; Schematic created using CircuitLab


It sounds like the isolation between the antennas isn't great. Maybe the antennas just aren't physically far enough apart, or their wires are too close together, resulting in crosstalk. 
Switching every few mS probably rules relays out. Here's a diode switcher you could use that only uses common components. The buffer amplifiers should provide approximately 50 ohm output impedance to reduce attenuation. It only needs one GPIO. You could also use jfets for switching but they're less common.