Can I capture the output of a started process without 'stealing' the output?

Lots of answers to other questions tell you that you can capture the output of a process using code looking something like this:

public async Task Run()
{
    await Task.Run(() =>
    {
        using (Process process = new Process()
        {
            StartInfo = new ProcessStartInfo
            {
                WindowStyle = ProcessWindowStyle.Normal,
                RedirectStandardOutput = true,
                RedirectStandardError = true,
                FileName = "cmd",
                Arguments = "path/filename.bat",
            }
        })
        {
            process.OutputDataReceived += Process_OutputDataReceived;
            process.ErrorDataReceived += Process_ErrorDataReceived;
            process.Start();
            process.BeginOutputReadLine();
            process.BeginErrorReadLine();

            process.WaitForExit();
        }
    });
}

private void Process_ErrorDataReceived(object sender, DataReceivedEventArgs e)
{
    if (e.Data != null)
    {
        log.Error(e.Data);
    }
}

private void Process_OutputDataReceived(object sender, DataReceivedEventArgs e)
{
    if (e.Data != null)
    {
        log.Info(e.Data);
    }
}

There are lots of variants which achieve the same thing more or less but they all redirect the standard output/errors - is there a way to capture them while still having them show in the cmd window?