How to hook up an FPGA to a component that has a different power supply?
The 20v voltage which you see across 2 chassis is high impedance sourced from mains. The schematics is a diamond shaped bridge fromed by 4 capacitors, real and parasitic. Top and bottom of bridge is mains. Horisontal of bridge is your voltmeter. Shoulders of bridge is this small ceramic UL certified capacitors, capacitive coupling of power cable to chassis, intercoil capacitance of power transformers. There is also an inductive background. The non-zero figures on voltmeter are caused by asymmetry.

If your equipment (power supplies) is generally trusty, you can tie grounds safely, so the few microamperes AC will be shorted, and it will resolve the problem.

But in your case you use something home brewed in combination with insulated device: modem is insulated by design with output transformer. So, no easy solution.

First class solution is creating lab grade ground using real earth and adding isolating transformer. Check the local electrical code as well. Only after this you can tie grounds.