var claimsIdentity = new ClaimsIdentity(new Claim[] { new Claim(ClaimTypes.Name, loginName) }, "Basic");
var claimsPrincipal = new ClaimsPrincipal(claimsIdentity);
await context.Authentication.SignInAsync(_cookieAuthOptions.AuthenticationScheme,
  claimsPrincipal,
  new AuthenticationProperties
  {
    IsPersistent = isPersistent,
    ExpiresUtc = DateTimeOffset.Now.Add(_cookieAuthOptions.ExpireTimeSpan)
  });

public IHttpActionResult GetAuthCookie(string loginName, bool isPersistent)
{
  var cookie = FormsAuthentication.GetAuthCookie(loginName, isPersistent);
  return Json(new { cookie.Name, cookie.Value, cookie.Expires });
}

public class UserServiceAgent
{
  private static readonly HttpClient _httpClient = new HttpClient();
  public static async Task<Cookie> GetAuthCookie(string loginName, bool isPersistent)
  {
    var response = await _httpClient.GetAsync(url);
    response.EnsureSuccessStatusCode();
    return await response.Content.ReadAsAsync<Cookie>();
  }
}

var cookie = await _userServiceAgent.GetAuthCookie(loginName, isPersistent);
var options = new CookieOptions()
{
  Domain = _cookieAuthOptions.CookieDomain,
  HttpOnly = true
};
if (cookie.Expires > DateTime.Now)
{
  options.Expires = cookie.Expires;
}
context.Response.Cookies.Append(cookie.Name, cookie.Value, options);