I was looking at a MFJ remote antenna tuner, and it is specified as being
able to match 12 to 1600 ohms.

What does that mean? At 12 ohms the SWR would be about 4.2:1 and at 1600
ohms a SWR of about 32:1.

If one started from 1600 ohms around a Smith chart, the circle would
intersect at the other side at about 0.64 ohms.

Likewise starting at 12 ohms and going halfway around the chart would end up
at 210 ohms, not 1600.

How does one use the specs if the antenna to be matched has a complex
impedance? Look at the internal tuner component ranges?