  The so called bipartite non-signaling boxes are systems whose statistics is
constrained solely by the principle of no instantaneous signaling between
distant locations. Such systems can exhibit much stronger correlations than
those admitted by quantum mechanics. Inspired by quantum logic approach of
Tylec and Ku\'{s}, J. Phys. A: Math. Theor. 48 (2015) 505303, we consider
non-signaling boxes with three inputs per party, and extend the set of
measurements with just a single {\it global} measurement - one that mimics
quantum two-party Bell measurement. We then show that this seemingly mild
extension completely rules out supraquantum correlations: the resulting system
admits precisely quantum mechanical correlations of two qubits. We also
consider non-maximally entangled measurements, obtaining interpolation between
quantum and full no-signaling theory. Our study paves a way to a general
programme of amending no-signaling theories with some measurements inherited
from quantum mechanics, leading to various interpolations between non-signaling
boxes and quantum mechanics.
