feat: declare workspace repository in installation config

backend/src/workspaces/registry.ts

@@ -11,6 +11,7 @@ import {
   GitWorkspaceRepository,
   WorkspaceRegistryError,
   WorkspaceRepositoryLock,
+  normalizeRepositoryIdentity,
   type GitStatus,
 } from "./git-repository.js";
 import {
@@ -492,6 +493,9 @@ export class WorkspaceRegistry {
     if (!active) throw safeError;
     return {
       branch: this.config.branch,
+      ...(this.config.remoteUrl
+        ? { repository: normalizeRepositoryIdentity(this.config.remoteUrl) }
+        : {}),
       head: active.head,
       ahead: 0,
       behind: 0,