Changing the base year of Real GDP from 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:
$$RGDP_{t, b2}  =RGDP_{t, b1} \cdot p_{base1} /  p_{base2}$$