In any event, it'll certainly take some time, and it's not going to be a particularly pleasant process. The bottom line: always, always, always keep your passport on your person or in your personal item — one that will never be gate-checked.
Should you carry a copy of your passport when you travel?
Safeguard your passport. While overseas, carry copies of your passport ID page and foreign visa with you at all times. If your passport is lost or stolen, report it immediately to the nearest embassy or consulate, or at 1-877-487-2778.
How do I carry my passport when traveling internationally?
The very best way to keep your passport safe is to carry it on your person in an anti-theft pouch or money belt that is hidden under your clothes. Here is a selection of anti-theft money belts and wallets. Here is a selection of anti-theft travel purses with special anti-theft zippers.