output.log

System.ArgumentNullException: Value cannot be null.
Parameter name: obj
   at ExceptionStackKeeper.Program1.BarAsync[T](T obj) in C:\Users\Owner\Source\Repos\ExceptionStackKeeper\ExceptionStackKeeper\Program1.cs:line 41
   at ExceptionStackKeeper.Program1.<FooAsync>d__1.MoveNext() in C:\Users\Owner\Source\Repos\ExceptionStackKeeper\ExceptionStackKeeper\Program1.cs:line 35
System.ArgumentNullException: Value cannot be null.
Parameter name: obj
   at System.Runtime.ExceptionServices.ExceptionDispatchInfo.Throw()
   at ExceptionStackKeeper.Program2.Main(String[] args) in C:\Users\Owner\Source\Repos\ExceptionStackKeeper\ExceptionStackKeeper\Program2.cs:line 19
System.ArgumentNullException: Value cannot be null.
Parameter name: obj
   at ExceptionStackKeeper.Program2.Main(String[] args) in C:\Users\Owner\Source\Repos\ExceptionStackKeeper\ExceptionStackKeeper\Program2.cs:line 27

Program1.cs

using System;
using System.IO;
using System.Runtime.ExceptionServices;
using System.Runtime.Serialization.Formatters.Binary;
using System.Threading.Tasks;

namespace ExceptionStackKeeper
{
    class Program1
    {
        static void Main(string[] args)
        {
            try
            {
                Task.Run(() => FooAsync()).Wait();
            }
            catch (AggregateException ex)
            {
                var formatter = new BinaryFormatter();
                using (var stream = new MemoryStream())
                {
                    formatter.Serialize(stream, ex);
                    Console.Write("new byte[] {");
                    foreach (var b in stream.ToArray())
                    {
                        Console.Write(" " + b + ",");
                    }
                    Console.WriteLine(" };");
                }
            }
            Console.ReadLine();
        }

        private static async Task FooAsync()
        {
            await BarAsync<string>(null);
        }

        private static Task<T> BarAsync<T>(T obj)
        {
            if (object.Equals(obj, default(T)))
                throw new ArgumentNullException(nameof(obj));

            return Task.FromResult(obj);
        }
    }
}

Program2.cs

using System;
using System.IO;
using System.Runtime.ExceptionServices;
using System.Runtime.Serialization.Formatters.Binary;
using System.Threading.Tasks;

namespace ExceptionStackKeeper
{
    class Program2
    {
        static void Main(string[] args)
        {
            using (var data = new MemoryStream(/* put Program1 output here */))
            {
                var formatter = new BinaryFormatter();
                var ex = (AggregateException)formatter.Deserialize(data);
                Console.WriteLine(ex.InnerException.ToString());
                try
                {
                    ExceptionDispatchInfo.Capture(ex.InnerException).Throw();
                }
                catch (ArgumentNullException ex1)
                {
                    Console.WriteLine(ex1.ToString());
                }
                try
                {
                    throw ex.InnerException;
                }
                catch (ArgumentNullException ex1)
                {
                    Console.WriteLine(ex1.ToString());
                }
            }
            Console.ReadLine();
        }
    }
}