Hi @Brent_W,
Datums and coordinate systems are tricky things to get a good handle on, but a short answer:
Coordinate systems are based on a particular model of the earth (datum), so a coordinate system derived from one datum needs to be appropriately projected if it’s being related to a different coordinate system based on a different datum.
ESRI puts out some reasonably good help files that explain why this matters, but to start… http://desktop.arcgis.com/en/arcmap/10.3/guide-books/map-projections/datums.htm
I don’t know if RTKlib can handle different CORS and Reach coordinate systems on-the-fly, but you might try using the web tool or downloading the Java application for NOAA’s Vdatum tool. https://vdatum.noaa.gov/
Theoretically, you should be able to convert your Reach data to the CORS coordinate system, or vice-versa. Not sure what steps you’d need to take to preserve formatting, but worth a try?