The use of convex relaxations has lately gained considerable interest in Power Systems. These relaxations play a major role in providing quality guarantees for non-convex optimization problems. For the Optimal Power Flow (OPF) problem, the semidefinite programming (SDP) relaxation is known to produce tight lower bounds. Unfortunately, SDP solvers still suffer from a lack of scalability. In this work, we introduce an exact reformulation of the SDP relaxation, formed by a set of polynomial constraints defined in the space of real variables. The new constraints can be seen as "cuts", strengthening weaker second-order cone relaxations, and can be generated in a lazy iterative fashion. The new formulation can be handled by standard nonlinear programming solvers, enjoying better stability and computational efficiency. This new approach benefits from recent results on tree-decomposition methods, reducing the dimension of the underlying SDP matrices. As a side result, we present a formulation of Kirchhoff's Voltage Law in the SDP space and reveal the existing link between these cycle constraints and the original SDP relaxation for three dimensional matrices. Preliminary results show a significant gain in computational efficiency compared to a standard SDP solver approach.