Changing the base year of Real GDP from 2001 Euros to 1990 Geary–Khamis dollars Estimating regional Gdp in Italy (1871-2001): 
sources, methodology, and results (Felice, E)

This article has Italian historical regional GDP figures, but they are in constant 2001 euros not in 1990 Geary–Khamis dollars where I can compare them more easily with other GDP figures. Would someone show me how you go about converting 2001 euros to 1990 Geary–Khamis dollars?
In general, real output is  calculated like this:
$$RGDP_{t, b} = NGDP_t \cdot p_{base} / p_t$$
Where $NGDP_t$ is nominal GDP at time t, $p_t$ is the price level in time $t$ and $p_{base}$ is the price level in the base period.

To covert RGDP between base periods, note that the ratio to a common year's $RGDP_t$ in two different base years is:
$$RGDP_{t, b1} / RGDP_{t, b2}  =  p_{base1} /  p_{base2}$$

You can rearrange this equality to solve for what you want:
$$RGDP_{t, b2}  =RGDP_{t, b1} \cdot p_{base1} /  p_{base2}$$