Electronic – Multiplexing a phase shift keyed signal

Yes, have a look at these AD analog switches. You can have SPST or SPDT switches, several of them with on resistances less than 1\$\Omega\$.

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I first suggested the ADG849, but Arik correctly points out that this is only limited to 5V operation. I had checked this, but I must have changed my mind and picked a different device at the last moment.

The ADG1436 is a better choice, also because it's a dual switch, so you only need one to switch Rx and Tx. It can operate at \$\pm\$15V, and switch signals to the rails. EIA-232 is \$\pm\$15V maximum, but chances are that you have 12V or, if you use something like a MAX232, only 10V.
The ADG1436 has an \$R_{ON}\$ of 2.6\$\Omega\$, and switches in about 100 nanoseconds. Contrary to a relay, also hardly requires any power: 0.001\$\mu\$A(!) at 25°C.

I would suggest that if you consider multiplexing with a ratio such that one can achieve acceptable brightness without needing the the instantaneous drive current to exceed the maximum allowable continuous operating current. If you don't, then you'll either have to add circuitry to ensure that the display shuts down when the multiplexing isn't happening properly, or else risk damaging the LEDs if a row or column gets stuck "on".

Given that constraint, depending upon how much brightness you need, you may find that somewhere between 2:1 and 8:1 is adequate. Suppose you decide on 4:1. In that case, I would suggest taking digits in groups of eight, and having each group of eight digits use one 16-bit shift register with current-controlled outputs for the display segments (eight to the segment and decimal wires of the first four displays, and eight to the other four) and an 8-bit shift register to select one of four common wires to drive. The spare outputs of the latter shifter could be used to do something like set the amount of current for the segment outputs [allowing for fading or flashing effects].

If you e.g. want some digits to show up as 25% brightness and the rest to repeatedly fade between 25% and full brightness, then for 1/4 of your multiplex cycles you should set the controller for full current and output the data for all the digits. For the other 3/4 of the cycles, you should set the controller for a variable amount of current, but blank all digits except those that should be fading. This approach has some limitations, but is often much easier than fancier PWM techniques.