I am not the most experienced when it comes to this setup but I see two ways that you could accomplish this.

Option 1 Custom Application Page

Create a page with no data for the News/Calendar in each site. Make the template pull in the contents using C# and Umbraco page queries. Sub pages would be accomplished by interrogating the URLs.

Option 2 Microsoft ARR

You might be able to tie the global lists section to a domain/port say localhost:8081 and use URL Rewrite + ARR to rewrite the site1.com/News to go to localhost:8081/news.

No one would be able to get directly to localhost except for on the server. Microsoft ARR would just reverse proxy traffic there. It is not a hard setup and does not require an IIS reboot. You just install, turn on the proxy and update the web.config.

ARR sounds complicated, but it isn't. Especially if you already understand the fundamentals of web.config and URL rewrite. It just allows you to URL rewrite outside of your app.

Microsoft ARR documentation

https://learn.microsoft.com/en-us/iis/extensions/url-rewrite-module/reverse-proxy-with-url-rewrite-v2-and-application-request-routing

For anyone else who finds this thread, I worked out a solution for this scenario which seems to work quite nicely.

1. Create an IContentFinder:

   public class SharedContentFinder : IContentFinder
   {
       private readonly IUmbracoContextAccessor _umbracoContextAccessor;
   
   
   public SharedContentFinder(IUmbracoContextAccessor umbracoContextAccessor)
   {
       _umbracoContextAccessor = umbracoContextAccessor;
   }
   
   
   public Task<bool> TryFindContent(IPublishedRequestBuilder contentRequest)
   {
       var path = contentRequest.Uri.GetAbsolutePathDecoded();
   
   
       if (!_umbracoContextAccessor.TryGetUmbracoContext(out var umbracoContext))
       {
           return Task.FromResult(false);
       }
   
   
       var sharedContent = umbracoContext.Content.GetByRoute(path);
   
   
       if (sharedContent is null)
       {
           // If not found, let another IContentFinder in the collection try.
           return Task.FromResult(false);
       }
   
   
       // If content is found, then render that node
       contentRequest.SetPublishedContent(sharedContent);
       return Task.FromResult(true);
   }
   
   
   }
   

2. Create an extension method for adding it to the builder:

   public static class SharedContentBuilderExtensions
   {
       public static IUmbracoBuilder AddSharedContentFinder(this IUmbracoBuilder builder)
       {
           builder.ContentFinders().Append<SharedContentFinder>();
           return builder;
       }
   }
   

3. Add it to your builder:

   builder.CreateUmbracoBuilder()
       .AddBackOffice()
       .AddWebsite()
       .AddDeliveryApi()
       .AddComposers()
       .AddSharedContentFinder()
       .Build();
   

4. Anywhere your views access the root node using something like this you will get errors as the chosen node is not a direct descendent of your root node:

       HomePage rootNode = (Model.Root() as HomePage);
   

.. so you'll instead need to use something like this:

        var domain = DomainService.GetByName(Context.Request.Host.Value);
        HomePage rootNode = (Umbraco.Content(domain.RootContentId) as HomePage);

is working on a reply...