Firms connected to your tour are outside companies (a management firm, a business manager, a booking agency, a travel agent) that asked to see part of your tour and that you approved. Nothing from your tour reaches a firm until you say yes.
APPROVING: a request shows on your Home Board as "A firm wants access to your tour". It lists exactly what they will see. Pick which tour, then How far back can they see? Full tour history (they see the whole tour, including shows already settled) or From today on (only shows and settlements from today forward). Full history is pre-picked for a management firm or business manager, because their job is the whole tour; you can change it before you tap Approve.
WHEN A FIRM ASKS FOR PAST HISTORY: a firm you connected "from today on" can ask to see earlier shows. That shows up on your Home Board as "asks to see your past tour history": Share full history or Not now. Saying not now changes nothing about what they already see. You can also open history any time yourself: Settings > Connections > Connected accounts, then Share more of your history.
STOPPING SHARING: Settings > Connections > Connected accounts, then Revoke. ATOM asks you to confirm first. They lose access right away: dates, day sheets, crew and new settlements. A management firm or business manager keeps the settled shows it already had, as read-only records, the way a manager keeps their own commission history. The client stays on their roster, marked as no longer connected.