The problem of local connectivity of the Mandelbrot set goes back to the 80s, and is now closely linked to the Renormalization Theory of quadratic polynomials. A key task is to establish a priori bounds (compactness) for the quadratic-like renormalization operator. Working in the near degenerate regime, we prove such bounds for maps with real combinatorics. As a consequence, real combinatorial classes are singletons on the real line. We also obtain a uniform control of the shapes of real-symmetric copies of the Mandelbrot set, as well as their universal scaling properties. Joint work in progress with Jeremy Kahn and Misha Lyubich.