I'm not sure I'm following 100%, but here is the answer to what I think you are asking:
Your line of thinking was common before the financial crisis. Most people assumed that 6M LIBOR was roughly equal to 3m LIBOR compounded to 6m (using fixing and 3m implied forward), and in fact swap curves were constructed using multiple fixings at the short end.
However, LIBOR is an unsecured lending rate between tier 1 banks, which means a 6m loan has twice the credit risk as a 3m loan. In late 2008, this basis blew out (you can see prior, it was fairly close to zero) and has stayed consistently above 0 since.
Also, you shouldn't use the direct ED implied rates- you need to imply a convexity adjustment. There are also stub adjustments you need to make. EDU0 represents a 3m loan starting on 9/16. That is ~3.5 months from now and not the exact 3m forward rate.
FRAs are more straightforward as they don't need a convexity or stub adjustment. I used them in the plot below, and you can see the basis is more consistently positive vs. using unadjusted ED rates, or compounding the 3m fixing twice which doesn't take curve shape into account.