Hmmm, this is tough one, so naturally it is interesting. One application I saw recently simply chose to use a small motor with very low torque with no other controls other than a limited power supply. When the motor takes up the slack of the tape it just stalls and does nothing except keep the tension on. You would want to pick a very small motor and put a current limiter in line with it so that while it is stalling it doesn't overheat or anything. This sounds kind of crude though, so not my first choice, but it could be quite simple.
Personally I would find a mechanical solution using an additional spring tensioner of some sort to take up additional slack.