Imagine that in an economy we only produced a toy and that cost us 10 dollars. We don’t sell this toy in year 1. Then this goes under Investment as +10 and GDP is +10. But say the only thing we do in year 2 is sell this toy for a loss at 8 dollars. Then investment is now -10 but consumption is +8 so GDP in year 2 is -2.
Is this a valid example of negative GDP?
Yes, theoretically, GDP could be negative, and you have described a pathological case where that happens. Gross value added in your example over the two periods is still positive, but the particular distribution of net value added over the two periods makes GDP in the second period negative.
You could think about this as a two-stage production process. In the first step, taking place in period 1, there is net value added of $10 in production of toys not yet delivered to retailers. Thus, GDP in period 1 is 10 dollars. In period 2, the retailers can only sell these goods for 8 dollars, so their net value added is -2 dollars. If this is the only economic activity, GDP in period 2 would be -2 dollars.
Sidenote: You have essentially described a pathological example of the inventory change example outlined in the BEA's NIPA handbook, Chapter 7 (https://www.bea.gov/resources/methodologies/nipa-handbook/pdf/chapter-07.pdf)
