How to get a visa reissued in France after having lost my passport?

I don't know if there is an established procedure but, promoting my comment to an answer, note that the relevant department for all things related to foreign nationals staying in France is the “service des étrangers” of the “préfecture” of the département where you currently are or, in Paris, the “préfecture de police”. That's who you should be trying to contact (ideally have a French-speaking friend contact them, they are not always the easiest to deal with).

Also, do report the theft to the police (go there with a French speaking person, never alone), it would be useful to have a police report and any document establishing what you are doing in France and for how long for the exit check or if there is a control before that.

1. You may not need the visa to continue staying, unless some immigration officials catch hold of you somewhere and ask to prove status. I doubt if you should face issues when exiting/ flying out.

2. Whoever/ whichever organization that sponsored your visa, will have legal counsel who can help you figure out your next steps/ action plan should be; as they had helped you with paperwork that got you your visa - any restamping or reapplication should be with their help

3. Not sure how risky it would be, but if you were to email/ go to a local French foreign services office and ask them what you should do, you may get a response.

   • Maybe ask a local french friend to do the above for you - less risky

   • Maybe ask a friend / family to do it at a Consulate local to them

I'm not sure you can get your Visa reissued in France. Quoting the Apply for Visa to France in India and Nepal site, question 10:

What do I have to do if I have lost or misplaced my Indian passport with a valid Schengen visa?

You are strongly recommended to keep photocopies of your passport’s data pages and of the Schengen visa. A visa is a valuable document. Inform The Embassy / The Consulate from where your visa was issued immediately if you have lost your passport. You must apply for a new visa by submitting all documents and payment of visa fees. You must submit a copy of the Police First Incident Report (FIR) with your application. It is preferable to also submit a copy of the visa and the lost passport for ready reference of the Visa Officer.

Have you made copies of your passport and Schengen visa before it was stolen?