First tab, for the demo to work right out of the box this sketch has to be called "SecondWindowTest"

/**
 * Demo on how to spawn a second sketch from Processing
 * Facilitated by ControlP5, but not necessary for this purpose
 * by CMB, 2016
 *
 */



import controlP5.*;

ControlP5 cp5;

//Reference to the secondary sketch so we can call its methods and access its variables from outside
SecondSketch second;

//This value will be accessed by the other sketch
int backgroundValue = 0;

public void settings()
{
  size(800, 600);
}

public void setup()
{
  surface.setResizable(true);

  cp5 = new ControlP5(this);

  cp5.addButton("secondWindow")
    .setPosition(20, 20)
    .setSize(125, 25)
    .setLabel("Second Window")
    .getCaptionLabel().alignY(CENTER).alignX(CENTER);

  cp5.addTextfield("textField")
    .setSize(125, 25)
    .setPosition(20, 75);
}

public void draw()
{
  background(backgroundValue);
}

public void secondWindow()
{
  //Launch the other sketch!

  //This structure assumes you only want one instance of the other sketch
  //If that's not what you want simply call the constructor to launch a new, independent instance

  //if the other sketch is closed, start a new one
  if (second== null) second = new SecondSketch(this);
  else
  {
    //if it's already running, bring it to the front
    second.moveToTop();
  }
}

public void textField(String input)
{
  //Send some text to the other sketch
  if (second != null)
  {
    second.someText = input;
  }
}

Second tab

//Code in this class like in any other sketch
//However, this class can't have any nested classes like a regular sketch:
//if you want to use another class inside this secondary sketch and draw in it,
//you gotta pass it a reference to this class and draw on its internal PGraphics "g"
class SecondSketch extends PApplet
{
  //A reference to the main sketch
  //notice how the file name of the main sketch is important as that becomes the name of the class upon compiling
  SecondWindowTest parent;

  ControlP5 gui;

  //This String will be sent to us from the main sketch
  String someText= "";

  //The constructor will take care of everything
  public SecondSketch(SecondWindowTest parent)
  {
    //store a reference to the first sketch so we can do things with it
    this.parent = parent;

    //This will actually launch the new sketch
    runSketch(new String[] {
      "SecondSketch"  //must match the name of this class
      }
      , this);  //the second argument makes sure this sketch is created instead of a brand new one...
  }

  public void settings()
  {
    size(700, 500);
  }

  public void setup()
  {

    super.surface.setResizable(true);

    //give it some GUI to interact with
    gui = new ControlP5(this);
    gui.addSlider("mainBackground")
      .setRange(0, 255)
      .setPosition(20, 50)
      .setLabel("Main sketch background colour")
      .setSize(125, 25);
  }

  public void draw()
  {
    background(50);
    fill(255);

    //Display the text that can be altered from the main sketch
    text(someText, 20, 20);
  }

  //Without this, the main sketch would close if the user clicked on the second sketch' close button!
  //(it appears to work funky with the escape button though, beware)
  public void exit()
  {
    parent.second = null;
    dispose();
  }

  //Callback from ControlP5
  public void mainBackground(float value)
  {
    //Change main sketch' background value
    parent.backgroundValue = (int) value;
  }

  //Little hack to bring the frame to the top
  public void moveToTop()
  {
    surface.setAlwaysOnTop(true);
    surface.setAlwaysOnTop(false);
  }
}

Any suggestions?

This doesn't really deal with core or contributed libraries, it's not really a hardware thing, and this seems like the most general place to put it, but apologies if there is a better place this should be.