The most common analog medium to mix to is 1/4" or 1/2" (reel to reel ) tape. Don't use the "home" equipment from Sony, RCA etc. as these use slower tape speeds (3-1/4 or 7-1/2 ips) and usually narrower track widths (you could flip the tape over, like a cassette - so it actually stored 4 tracks on the tape, 2 one direction and 2 the other). A professional half-track machine from Studer, Ampex, Otari, or even a semi-pro like Tascam at 15 or 30 ips will give the sought after "analog warmth". Beware, though, as a lot of the machines on the used market will need expensive work to get them operating right. A good analog machine must be kept up - heads must be demagnetized and periodically replaced or re-lapped; bias, azimuth, Record & reproduce levels and EQ must all be properly set on a regular basis, the pinch roller must be treated and the entire tape path kept clean and properly aligned. A demagnetizer and an alignment tape for your specific model machine are must haves, as is access to a technician who can handle the maintenance and repairs that you can't. This is not a "buy it, set it up & turn it on when you need it" proposition. Unless you're willing to spend the time, money and effort, just skip the analog tape and get something like the Fatso or the HEDD to simulate it for you.
Scott