Found it!

https://learn.microsoft.com/en-us/aspnet/core/blazor/security/?view=aspnetcore-9.0&tabs=visual-studio#manage-authentication-state-in-blazor-web-apps

Add this to the server project's Program.cs:

builder.Services.AddRazorComponents()
    .AddInteractiveWebAssemblyComponents()
    .AddAuthenticationStateSerialization(
        options => options.SerializeAllClaims = true);

Add this to the client project's Program.cs:

builder.Services.AddAuthorizationCore();
builder.Services.AddCascadingAuthenticationState();
builder.Services.AddAuthenticationStateDeserialization();

Wrap the Router component (in Routes.razor) in a CascadingAuthenticationState component. Looks like this:

@using Microsoft.AspNetCore.Components.Authorization
<CascadingAuthenticationState>
    <Router AppAssembly="typeof(Program).Assembly">
        <Found Context="routeData">
            <RouteView RouteData="routeData" DefaultLayout="typeof(Layout.MainLayout)"/>
            <FocusOnNavigate RouteData="routeData" Selector="h1"/>
        </Found>
    </Router>
</CascadingAuthenticationState>

Test page:

@page "/test"
@using Microsoft.AspNetCore.Components.Authorization
@inject AuthenticationStateProvider AuthenticationStateProvider

<h3>User Claims</h3>

@if (userName is null)
{
    <p>Loading...</p>
}
else
{
    <p>Hello, @userName!</p>
    <ul>
        @foreach (var claim in userClaims)
        {
            <li>@claim.Type: @claim.Value</li>
        }
    </ul>
}

@code {
    private string? userName;
    private IEnumerable<System.Security.Claims.Claim> userClaims = Enumerable.Empty<System.Security.Claims.Claim>();

    protected override async Task OnInitializedAsync()
    {
        // Get the current authentication state
        var authState = await AuthenticationStateProvider.GetAuthenticationStateAsync();
        var user = authState.User;

        if (user.Identity is not null && user.Identity.IsAuthenticated)
        {
            userName = user.Identity.Name;
            userClaims = user.Claims;
        }
        else
        {
            userName = null;
            userClaims = Enumerable.Empty<System.Security.Claims.Claim>();
        }
    }
}