processredirect.ps1

$psi = [System.Diagnostics.ProcessStartInfo]@{
    UseShellExecute = $false
    RedirectStandardError = $true
    RedirectStandardOutput = $true
    FileName = 'powershell.exe'
    Arguments = '$host.UI.WriteLine(''stdout''); $host.UI.WriteErrorLine(''stderr''); Start-Sleep -Seconds 5'
}
$proc = [System.Diagnostics.Process]::new()
$proc.StartInfo = $psi

Register-ObjectEvent -InputObject $proc -EventName OutputDataReceived -SourceIdentifier Process.Stdout -Action {
    "STDOUT: $($EventArgs.Data)"
} | Out-Null
Register-ObjectEvent -InputObject $proc -EventName ErrorDataReceived -SourceIdentifier Process.Stderr -Action {
    "STDERR: $($EventArgs.Data)"
} | Out-Null
Register-ObjectEvent -InputObject $proc -EventName Exited -SourceIdentifier Process.Exited

try {
    $proc.Start() | Out-Null
    $proc.BeginOutputReadLine()
    $proc.BeginErrorReadLine()

    Wait-Event -SourceIdentifier Process.Exited | Remove-Event
}
finally {
    Unregister-Event -SourceIdentifier Process.Stdout
    Unregister-Event -SourceIdentifier Process.Stderr
    Unregister-Event -SourceIdentifier Process.Exited
}