The structure of a public bid is most of the time the same. This means that the documentation asked to be provided is also very similar from bid to bid. Here is a small list of things that you should have available and up to date (in french of cource). This would be the overall documentation mostly excluding bid specific documents.
DUME (ESPD in English): The European single procurement document (ESPD) is a self-declaration form used in public procurement procedures. Before the introduction of the ESPD, companies had to submit various documents proving that they can participate in a procurement procedure (e.g. on having paid taxes, on not having been convicted of criminal activity). The ESPD replaces the DC1, DC2, DC4 forms that you would have to fill otherwise. This is mandatory and must be filled in french.
The DUME will require your company annual turnover for the last three years (in the event of a lower financial year or more than 12 months, it is reduced to 12 months pro rata temporis). This must be up to date for each bid.
A portfolio of your work from the last 3 years. This has been asked in each bid we have encountered. This should be one of the first things to do when entering the french public market. Create/translate a portfolio of your last 3 year projects and automatically add in any additional projects done after that. Each example should be then accommodated the bid you are trying to win to show how this proves that you are the right fit for the job.
A short presentation\resume with photos of each person working on your project. it must be clearly stated which persons do speak french and will be responsible for communication. It is not always asked but from our experience it should be provided nonetheless. It will show the bidders who they will be dealing with and might give more confidence that you can manage the french market.
An electronic signature - Not mandatory by law during the application phase but in practice it is sometimes applicable if required in the bidding documents. It is mandatory for the contract signature if you win. According to french law all eIDAS regulated electronic signatures should do the trick. The reality is more complicated, not all foreign electronic signatures are working on french bidding platforms. We recommend testing your signatures beforehand. Bidding platforms fortunately provide testing options. If the signature is not working, then you should contact support and start fighting for your rights.
The last general advice is to always go the extra mile. As a foreigner you are fighting an uphill battle. The more you provide, the greater chance there is to break the cultural wall.