Is there any way to force audio through the speakers when headphones are plugged in?

The answer turned out to be the following, with tips from Android - Getting audio to play through earpiece

audioManager.setMode(AudioManager.MODE_IN_CALL);
audioManager.setSpeakerphoneOn(true);

Ignore setRouting, it does nothing in APK > 10. Ignore comments about setMode. Ignore comments about setWiredHeadsetOn. Make sure you have MODIFY_AUDIO permissions.

Thanks to some comments here, and after some search I finally found out how to do this.

My code below is called when I press a button. It sets up audio for activating speakers while calling, then calls the number.

Pay attention to the comments in my code regarding permissions.

buttonRep.setOnClickListener(new OnClickListener() {
    @Override
    public void onClick(View v) {
        //1) we need to detect once the call has been established
        //Note : this requires READ_PHONE_STATE permission in your manifest

        final TelephonyManager tm = (TelephonyManager) getSystemService(Context.TELEPHONY_SERVICE);

        tm.listen(new PhoneStateListener()
        {
            @SuppressWarnings("unused")
            @Override
            public void onCallStateChanged (int state, String incomingNumber)
            {
                if(state == TelephonyManager.CALL_STATE_OFFHOOK) //Here we are
                {
                    //2) we switch sound to speakers
                    //Note : this requires MODIFY_AUDIO_SETTINGS permission in your manifest
                    AudioManager am = (AudioManager) getSystemService(Context.AUDIO_SERVICE);
                    am.setMode(AudioManager.MODE_IN_CALL);
                    am.setSpeakerphoneOn(true);
                    int i = state;

                    //3) we don't need this call listener anymore so we "destroy it"
                    tm.listen(this, PhoneStateListener.LISTEN_NONE);
                }
            }
        }

        , PhoneStateListener.LISTEN_CALL_STATE);

        //4) we are ready to place our call  now
        //Note : this requires CALL_PHONE permission in your manifest
        Intent action = new Intent();
        action.setAction(Intent.ACTION_CALL);
        action.setData(Uri.parse("tel:660")); //660 is my answering-machine number :)
        MainActivity.this.startActivity(action);
    }
});

Try this:

    AudioManager mAudioManager = (AudioManager)getSystemService(Context.AUDIO_SERVICE);

    IntentFilter iFilter = new IntentFilter(Intent.ACTION_HEADSET_PLUG);
    Intent iStatus = getApplicationContext().registerReceiver(null, iFilter);
    boolean isHeadsetOn = false;

    if (iStatus != null) {
        isHeadsetOn = iStatus.getIntExtra("state", 0) == 1;
    }
    if(mAudioManager.isWiredHeadsetOn() || isHeadsetOn)
    {
        //When headphones are plugged
        mAudioManager.setMode(AudioManager.MODE_CURRENT);
        mAudioManager.setSpeakerphoneOn(true);
    }else { 
        //When headphones are not plugged
        mAudioManager.setMode(AudioManager.MODE_IN_CALL);
        mAudioManager.setSpeakerphoneOn(true);
    }