How can nominal GDP increase even though real GDP falls?

Primarily this happens because of increase in prices.

Nominal GDP= GDP using current prices.

Real GDP= GDP that takes prices changes into account.

Let me give a very simple example, let's say:

In year 1, the country produced 10 computers for 10 dollars each. So GDP for year 1= $100

In year 2, the country only produced 9 computers for 15 dollars each. So GDP for year 2 = $135 (9x15)

In year 2,the nominal GDP has increased from $100 to $135. However, we measure real GDP using a base year, in this case year 1, so we use the price of year 1 to find the real GDP for year 2.

Using prices of year 1 we have: 9 computers x $10 each = $90 of real GDP.

Finally, you see that even nominal GDP for year 2 was $135, the real GDP was $90.