Authentication via Web API in ASP .NET Core 9 Razor Pages web app (with Identity)

[bairog]

I have an ASP .NET Core 9 Razor Pages web app which uses Identity. I've created it from a template, attached my custom DbContext and scaffolded Identity/Pages/Account/Login page. And I've tried to add Web API to the same project to be able to have other consumers (iOS/Android applications, PHP/Go clients, etc.). With minimal API syntax I've implemented LoginUser:

app.MapGet("/api/LoginUser", async (string username, string password, SignInManager<IdentityUser> signInManager, IHttpContextAccessor httpContextAccessor) =>
{
    var result = await signInManager.PasswordSignInAsync(username, password, false, false);

    //IsLoggedIn is true here
    var IsLoggedIn = httpContextAccessor?.HttpContext?.User?.Identity?.IsAuthenticated;
    if (result.Succeeded)
    {
        return Results.Ok();
    }
    return Results.BadRequest($"SignInResult: {result}");
});

And now on posting a login form I perform an API call via HttpClient like that:

public async Task<IActionResult> OnPostAsync(string returnUrl = null)
{
    returnUrl ??= Url.Content("~/");

    using (var client = new HttpClient())
    {
        var url = $"{Request.Scheme}://{Request.Host}{Request.PathBase}/api/LoginUser?username={Input.Email}&password={Input.Password}";
        var response = await client.GetAsync(url);

        switch (response.StatusCode)
        {
            case System.Net.HttpStatusCode.OK:
                //But IsLoggedIn becomes FALSE here
                var IsLoggedIn = _httpContextAccessor?.HttpContext?.User?.Identity?.IsAuthenticated;

                return Redirect(returnUrl);
                break;

            case System.Net.HttpStatusCode.BadRequest:
                var errorText = response.Content.ReadAsStringAsync().Result;
                ModelState.AddModelError(string.Empty, errorText);
                return Page();
                break;
        }
    }
    // If we got this far, something failed, redisplay form
    return Page();
}

Problem here is that from PageModel point of view user remains not authenticated even after rage refresh.
If for example I navigate in browser to URL with a proper query string (https://localhost:7210/api/[email protected]&password=!TestPassword123) user becomes logged in after page refresh.

So how to login from a Razor Page via Web API in a correct way?
My sample repo to reproduce that (login with [email protected] and !TestPassword123).