As the name suggests, hybrid chess is a cross between over-the-board and online chess. The rules are given in the FIDE Online Chess Regulations document.
6.1.3 “Hybrid chess” competition is an event where all players are physically supervised by an arbiter, while they play online (see Part III b)
Basically such a competition involves players at different controlled "venues" playing online games against each other while under surveillance from cameras and arbiters (no spectators allowed to be physically present) via controlled internet access.
16.1. The Chief Organiser designates the playing venues for the competition. Each playing venue is under the control of a Local Organiser.
16.2 Each Local Organiser is required to provide a playing venue suitable to host a hybrid chess competition. The ‘playing venue’ is defined as the ‘playing area’, rest rooms, toilets, refreshment area, area set aside for smoking and other places as designated by the arbiter. The playing area is defined as the place where the games of a competition are played. Only players and arbiter are allowed access to the playing area.
16.3 Each playing venue must be monitored by cameras.
16.4 In each playing venue, the fair play measures should be applied in accordance with the FIDE AntiCheating Regulations and FIDE Anti-Cheating Protection Measures. Unless authorised by the arbiter, it is forbidden for anybody to use a mobile phone or any kind of communication device in the playing venue or any contiguous area designated by the arbiter.
16.5 Each Local Organiser is responsible for providing an Internet connection in the playing venue. Players are not responsible for their connections to HIP and to a communication system (if required by the competition regulations), unless the competition regulations say otherwise.
16.6 In each venue, electronic devices used for conducting online games (playing devices) are provided by the Local Organiser, unless the competition regulations say otherwise.
16.7 During the game, each player shall have access on his/her playing device to a virtual chessboard and any software required for this purpose. No other website, application or software can be open on the playing device. The only exception may be a (video-) communication system, if required by competition regulations.
16.8 At least two arbiters will be appointed for each playing venue: a Local Chief Arbiter (LCA) and a Local Technical Assistant (LTA).
16.9 The total number of arbiters required in each playing venue will vary depending on the kind of event, on the system of the games, on the number of participants and on the importance of the event.
16.10 If the playing zone allows players to move pieces in contradiction with Article 3.3 (illegal moves), the competition regulations must specify how to deal with such irregularities.
16.11 The competition regulations shall specify a default time in advance. If the default time is not specified, then it is zero. If the competition regulations specify that the default time is not zero and if neither player is present initially, White shall lose all the time that elapses until he/she arrives, unless the competition regulations specify or the arbiter decides otherwise.
16.12 The playing zone must record the offer of a draw next to the player’s move when the draw is offered.