Assumptions are being made. Is your adaptor rs232
or rs422? Wire accordingly, but it should be 422 as that is nmea 0183
signaling - I believe without double checking.
422 is differential, connect both - & + wires if that is the adaptor type. If it is an rs232
adaptor, you might just might, get away with connecting the 2 data - to ground. In all cases, connect ground from the device to the adaptor as it is likely passed through to the pc's USB.
The pc and rs232 are unrelated as you are using USB input. There hasn't been an rs232 port on a computer for at least 10, maybe 15 years.
Sorry if this is too much info.
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