Continuing troubleshooting steps, if this is possible via any of the ODBC, ADO, or direct, connectors you will be able to do it via the Tools → SQL command in the menu bar in Base. For the HSQLDB internal, the command is:
ALTER TABLE "Table1"
ADD CONSTRAINT "my_key" UNIQUE ("fld1")
I would guess that might work on your MS ACCDB backend, based on what is here and here although I cannot replicate because I also do not have Access.
That said, I suspect that Access would sneak in a primary key when done through ythat system and even though this command will add a UNIQUE constraint successfully, it will not allow you to enter new data through the Base front-end, which peterwt also appears to confirm is a limitation of the Base interface, although a mild one considering that adding a PRIMARY key is easy to do.