This subject has come up a few times, and some of the board experts had a few ideas to try. The simple one was just to get some decent 1:1 input iron to put in front of the converters. Not quite tape saturation, but some good color, perhaps.
There's also this circuit I found some time ago:
http://www.261.gr/limiters.html
If you scroll down to Figure 8 at the above link, they describe a diode-based soft clipper that supposedly introduces a tape-saturation effect. It uses a mix of silicon and germanium diodes.
I also found this little snippet of description on the Presonus site, for their Eureka channel strip:
The saturation circuit works by adjusting the drain current on the FET amplifier, and is intended to simulate the effects of tape saturation and tube "warmth."
Possibly enough there to play around with.
One other idea I had was to cannabalize a open-reel tape deck to record your signal to, then take the out from the playback head immediately to your computer. This would introduce latency, but as long as the latency is the same for each track, you can compensate for it. The disadvantages would be, one, potentially ruining a perfectly good tape deck, and two, a limit on the number of tracks you could process at a time, since you'd want to use a cheaper deck for this.
There are lots of ideas out there.
EDIT: I just realized this is my first post in the Drawing Board. Go me!! :wink: