Whenever I hear about Phillpe's, I can't help but bring up the Eastside Market on Alpine Street ("sandwiched" between Dodger Stadium and Chinatown). Although they don't serve the classic Dip that you'd enjoy at Phillipe's-- they serve up an awesome roast beef/pastrami concoction packed full of meat, cheese, peppers, and a juicy red sauce.
They have a boatload of other traditional, old world, L.A.-style Italian delicatessen fare too. Including a "just like mama made" minestrone, chock full of fresh veggies and pasta.
Be forewarned though, the place is popular. If you happen to stop by at lunch time, be prepared to stand in line behind L.A.'s finest and bravest... it's cop & firefighter central around noon. But it's well worth the extra wait (and a bargain too!)