We use nuclear magnetic resonance to map the complete low-temperature phase diagram of the antiferromagnetic Ising-like spin-chain system BaCo2V2O8 as a function of the magnetic field applied along the chains. In contrast to the predicted crossover from the longitudinal incommensurate phase to the transverse antiferromagnetic phase, we find a sequence of three magnetically ordered phases between the critical fields 3.8 T and 22.8 T. Their origin is traced to the giant magnetic-field dependence of the total effective coupling between spin chains, extracted to vary by a factor of 24. We explain this novel phenomenon as emerging from the combination of nontrivially coupled spin chains and incommensurate spin fluctuations in the chains treated as Tomonaga-Luttinger liquids.PACS numbers: 75.10. Pq, 75.30.Kz, 71.10.Pm, The study of emergent phenomena in interacting quantum systems is at the heart of condensed-matter physics. Interacting fermions confined to one dimension (1D) emerge in a quantum-critical state with non-particle-like excitations, whose low-energy description is known as the Tomonaga-Luttinger liquid (TLL) [1]. As any correlation function adopts a universal form, insensitive to the microscopic details, the TLL description applies to a wide range of systems, like 1D metals [2], edge states of quantum Hall effect [3], quantum wires [4], carbon nanotubes [5] and 1D arrays of atoms on surfaces [6] or in optical traps [7]. The simplest and experimentally most accessible TLLs are realized in 1D quantum antiferromagnets hosting spin chains or ladders, which can be mapped onto interacting spinless fermions [8]. In particular, two spin-ladder systems, (C 5 H 12 N) 2 CuBr 4 (BPCB) [9-13] and (C 7 H 10 N) 2 CuBr 4 (DIMPY) [14][15][16][17][18][19], allowed to confirm the predicted correlation functions not only in form, but also quantitatively as a function of the magnetic field, which controls the Fermi level [1].While isolated TLLs cannot order because of strong quantum fluctuations, a weak coupling between TLLs leads at low temperatures to the 3D ordered state, which inherits the properties of the dominant fluctuation mode.As the Fermi surface in a TLL is reduced to two points, k F and −k F , fermionic fluctuations can only occur at the wavevectors q = 0 and q = 2k F [1]. In antiferromagnetic spin chains or ladders in a magnetic field, the corresponding spin fluctuations are transverse (i.e., involving spin components perpendicular to the field) antiferromagnetic, at the antiferromagnetically shifted wavevector q = π, and longitudinal (i.e., involving spin components along the field) incommensurate at the incommensurate wavevector q = 2k F , respectively [1]. For the Heisenberg exchange between spins, the transverse fluctuations dominate and a weakly coupled system develops a transverse antiferromagnetic order at low temperatures [20,21], in the gapless region between the critical fields B c and B s , which correspond to the edges of the fermion band. Examples include BPCB [10,22] [24]. More interestingly, for t...