Yeah, MSW (Modified Sine Wave) inverters will do that sometimes, especially the cheap ones. There are several things to try. Try to determine where the noise is getting into the system, through the antenna or the power.
If the noise is entering through the antenna:
1. Ground the chassis of the radio by connecting a wire from a ground screw under the dash to a screw on the side of the radio.
2. The antenna cable may be picking up noise. Try re-routing the antenna cable, coiling it into a circle, or shielding it.
3. Try relocating the antenna to a different part of the vehicle (if possible).
If the noise is entering through the power leads:
1. Ground the chassis of the radio by connecting a wire from a ground screw under the dash to a screw on the side of the radio.
2. Try another source of power. The cleanest source of power is directly from the battery. If you connect directly to the battery (not to the fuse box), shielded cable is recommended to prevent noise pickup from the ignition system. Don't forget to use a 5 amp fuse right at the battery when connecting directly to the battery.
3. Try installing a noise filter on the power leads, as close as possible to the radio. Radio Shack sells several automotive noise filters, such as part number 270-051.
On the Inverter end:
1. Keep the inverter and inverter cables (both the battery cables and the AC cables) as far as possible from the CB and CB cables.
2. Connect a #12 wire from the ground lug on the inverter to the chassis of the vehicle. Keep the wire as short as possible.
3. If there is still interference, then unplug everything from the inverter and leave the inverter turned on. If you no longer get interference, then an AC line filter should help. Tripp-Lite ULTRABLOK428 is a real good one. This filter plugs directly into the outlet on the inverter. DO NOT use an extension cord between the filter and the inverter, plug the filter directly into the inverter. If desired, a power strip can then be plugged into the filter.
4. As a last resort, try connecting a #12 wire from the chassis of each appliance to the chassis of the vehicle. Keep each wire as short as possible.
PSW (Pure Sine Wave) inverters rarely cause CB and other interference.