"A tempo 80 piece in 4/4 will line perfectly with a tempo 80 piece in 12/8 (compound time signature)"

The way I was taught, 4/4 with a tempo shown normally means a quarter note (1/4) equals the tempo (80 BPM).
12/8 implies an eighth note equals 80 BPM.
In that case I don't see the quoted statement as being correct.

It may 'feel' like it, but technically ..
If I play a clip that lasts for (example) 2200 milliseconds, and it feels like two measures
If I divide 2200 milliseconds by 8 (for the two 4/4 measures)
Then take that same slice and divide it by 24 (for the two 12/8 measures) the two results would be very different numbers.

Like I said above, it may feel like it is the same when notated differently, but you're doing different math behind the scenes.
That said trained/experienced musicians do this all the time nowadays.

How would Tchaikovsky have notated it?

I do not work here, but the benefits are still awesome
Make your sound your own!