To request a duplicate copy of your tourist card ("Tarjeta Unica Migratoria" or sometimes "Tarjeta de Turismo"), you'll need to visit the extranjeria office.
The office is located at Eleuterio Ramirez N° 852 in centro. Note that they are open from 8 AM until 2 PM.
Also, make sure you bring a pen with you.
To get there via Metro, take the red line to Santa Lucía and exit Avda. Bernardo O'Higgins Súr. Walk west to San Fransisco and then go south 3 blocks to Eleuterio Ramirez. The building will be on your right.
Once you arrive, go to the table next to the entrance and pick up 2 forms:
- Replacement tourist card (looks the same as the one you were issued when you arrived in the country). Note that it has 3 carbon copies.
- A full-size form to request the replacement (I forget exactly what it's called, but it's the only other form on the table that mentions a tourist card).
Fill out the forms (you did remember to bring a pen, right?), and then walk straight to cubicle 10. There is no signage; you're just supposed to know that this is where you go to replace your tourist card.
There may be an informal line in front of this cubicle; be prepared to do a little negotiation. Also note that the PDI agent who processes your forms will probably only know Spanish.
Important: You do not need to take a number*, and you do not need to visit the cashier ("caja")**. There is no fee to replace your tourist card.
The whole process should take about 30 minutes.
* Source: four hours of my life that I will never get back.
** Source: another 30 minutes of my life that I also will never get back.