This is so silly. I hacked something together that worked for me, I hope it'll work for you too.
public class AuthRequestDialogFragment extends DialogFragment
{
    @InjectView(R.id.dauth_userinput)
    public EditText userinput;
    @InjectView(R.id.dauth_passinput)
    public EditText passinput;
    @OnClick(R.id.dauth_login)
    public void login(View view) {
        ((Callback) getTargetFragment()).login(userinput.getText().toString(), passinput.getText().toString());
        this.dismiss();
    }
    @OnClick(R.id.dauth_cancel)
    public void cancel(View view) {
        ((Callback) getTargetFragment()).cancel();
        this.dismiss();
    }
    public static interface Callback
    {
        public void login(String username, String password);
        public void cancel();
    }
    @Override
    public void onStart() {
        super.onStart();
        WindowManager wm = (WindowManager) getActivity().getSystemService(Context.WINDOW_SERVICE);
        Display display = wm.getDefaultDisplay();
        int width = display.getWidth();
        int height = display.getHeight();
        getDialog().getWindow().setLayout(width*2/3, height/5*2);
    }
    @Override
    public View onCreateView(LayoutInflater inflater, ViewGroup container, Bundle savedInstanceState)
    {
        View view = inflater.inflate(R.layout.dialog_authrequest, container);
        ButterKnife.inject(this, view);
        getDialog().setTitle("Authorization required");
        return view;
    }
}
And
<?xml version="1.0" encoding="utf-8"?>
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
                android:layout_width="wrap_content"
                android:layout_height="wrap_content">
    <TextView
        android:id="@+id/dauth_requsertext"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:text="The server requires a username and password."
        android:layout_alignParentTop="true"
        android:layout_alignParentStart="true"
        android:layout_alignParentLeft="true"
        android:layout_marginLeft="15dp"
        android:layout_marginStart="15dp"
        android:layout_marginTop="15dp"/>
    <RelativeLayout
        android:id="@+id/dauth_centercontainer"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:layout_centerInParent="true">
        <TextView
            android:id="@+id/dauth_usertext"
            android:layout_width="wrap_content"
            android:layout_height="wrap_content"
            android:text="Username"
            android:layout_alignParentTop="true"
            android:layout_alignParentStart="true"
            android:layout_alignParentLeft="true"
            android:layout_marginLeft="15dp"
            android:layout_marginStart="15dp"
            android:layout_marginTop="15dp"/>
        <TextView
            android:id="@+id/dauth_passtext"
            android:layout_width="wrap_content"
            android:layout_height="wrap_content"
            android:layout_marginTop="20dp"
            android:text="Password"
            android:layout_below="@+id/dauth_usertext"
            android:layout_alignLeft="@+id/dauth_usertext"
            android:layout_alignStart="@+id/dauth_usertext"/>
        <EditText
            android:layout_width="wrap_content"
            android:layout_height="wrap_content"
            android:id="@+id/dauth_userinput"
            android:ems="12"
            android:layout_alignBottom="@+id/dauth_usertext"
            android:layout_toRightOf="@id/dauth_usertext"
            android:layout_toEndOf="@id/dauth_usertext"/>
        <EditText
            android:layout_width="wrap_content"
            android:layout_height="wrap_content"
            android:id="@+id/dauth_passinput"
            android:layout_marginTop="20dp"
            android:inputType="textPassword"
            android:ems="12"
            android:layout_alignBottom="@+id/dauth_passtext"
            android:layout_toRightOf="@id/dauth_passtext"
            android:layout_toEndOf="@id/dauth_passtext"
            android:layout_alignLeft="@id/dauth_userinput"
            android:layout_alignStart="@id/dauth_userinput"/>
    </RelativeLayout>
    <Button
        android:id="@+id/dauth_cancel"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:text="Cancel"
        android:layout_alignParentBottom="true"
        android:layout_alignParentRight="true"
        android:layout_alignParentEnd="true"
        android:layout_marginBottom="15dp"
        android:layout_marginRight="15dp"
        android:layout_marginEnd="15dp"/>
    <Button
        android:id="@+id/dauth_login"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:text="Log In"
        android:layout_alignTop="@+id/dauth_cancel"
        android:layout_marginRight="15dp"
        android:layout_marginEnd="15dp"
        android:layout_toLeftOf="@+id/dauth_cancel"/>
</RelativeLayout>
And
public class WebViewFragment extends Fragment implements AuthRequestDialogFragment.Callback {
    @Override
    public void login(String username, String password) {
        Log.d(this.getClass().getName(), "Login");
        myWebViewClient.login(username, password);
    }
    @Override
    public void cancel() {
        Log.d(this.getClass().getName(), "Cancel");
        myWebViewClient.cancel();
    }
And the most important: 
private class MyWebViewClient extends WebViewClient {
    private WebView myView;
    private HttpAuthHandler httpAuthHandler;
    private String host;
    private String realm;
    @Override
    public void onReceivedHttpAuthRequest(WebView view, HttpAuthHandler handler, String host, String realm) {
        AuthRequestDialogFragment authRequestDialogFragment = new AuthRequestDialogFragment();
        FragmentManager fragmentManager = ((getActivity()).getSupportFragmentManager());
        authRequestDialogFragment.setTargetFragment(WebViewFragment.this, 0);
        authRequestDialogFragment.show(fragmentManager, "dialog");
        this.httpAuthHandler = handler;
        this.myView = view;
        this.host = host;
        this.realm = realm;
    }
    public void login(String username, String password) {
        httpAuthHandler.proceed(username, password);
        myView = null;
        httpAuthHandler = null;
        host = null;
        realm = null;
    }
    public void cancel() {
        super.onReceivedHttpAuthRequest(myView, httpAuthHandler, host, realm);
        myView = null;
        httpAuthHandler = null;
        host = null;
        realm = null;
    }
}
Uses dependency:
compile 'com.jakewharton:butterknife:6.0.0'