R1 on 1st, running on the pitch. B2 hits a low line drive to F6, who catches the ball. F6 then throws to F3, attempting to catch R1 off base before he can return to tag-up after the catch. R1 attempts to return to 1st but is obstructed by F4.
The ball reaches F3, who touches 1st and legally appeals before R1 can recover and return to the base. The umpire considers R1 would not have returned in time, even had the obstruction not occurred, so calls R1 out.
The offensive team coach then insists that the out should be overturned, claiming that no runner can be called out between the bases where he was obstructed.

The out stands
 

This is straight out of the Rule Book.

POE #36 - A runner leaving second base too soon on a fly ball is returning after the ball is caught and is obstructed between second base and third base. If the runner would not have made it back to second base prior to the throw arriving, the runner would remain out.

Except upon proper appeal of a base left too soon.

I know I'm merely restating in different words, but the "runner cannot be put out between the bases where the obstruction occurred" rule does not apply when another violation by the runner is being played on, such as leaving too soon on a caught fly.