What is Tracking Area Update Complete in LTE?

It is the EPS NAS message the UE sends after Tracking Area Update Accept when the accepted TAU branch requires explicit completion.

What should I inspect first in Tracking Area Update Complete?

Start by checking whether the accepted TAU branch really required completion, then correlate the message with the earlier accept and T3450 behavior.

Why is Tracking Area Update Complete important in troubleshooting?

Because it shows that the UE actually closed the accepted TAU branch instead of leaving the network waiting for completion.

What usually comes after Tracking Area Update Complete?

The UE continues in protected registered-state behavior, subject to the refreshed TAU context the network already accepted.