Look at so-called Parshin reciprocity law, for example Mazin' approach http://www.math.utoronto.ca/mmazin/thesis.pdf (cf. http://www.math.toronto.edu/askold/osaka.pdf)

I think that is exactly what you want.

Look at so-called Parshin reciprocity law, for example Mazin' approach http://www.math.utoronto.ca/mmazin/thesis.pdf

I guess that is exactly what you want.

If I am not mistaken, your version follows from the standard version of Weil reciprocity law for the functions $f/g,f/h$, in the spirit of logarithmic differentials (cf. http://www.math.toronto.edu/askold/osaka.pdf) If not, it is very strange.