  We present a comprehensive macroscopic thermodynamic study of the
quasi-one-dimensional (1D) $s = \tfrac{1}{2}$ frustrated spin-chain system
linarite. Susceptibility, magnetization, specific heat, magnetocaloric effect,
magnetostriction, and thermal-expansion measurements were performed to
characterize the magnetic phase diagram. In particular, for magnetic fields
along the b axis five different magnetic regions have been detected, some of
them exhibiting short-range-order effects. The experimental magnetic entropy
and magnetization are compared to a theoretical modelling of these quantities
using DMRG and TMRG approaches. Within the framework of a purely 1D isotropic
model Hamiltonian, only a qualitative agreement between theory and the
experimental data can be achieved. Instead, it is demonstrated that a
significant symmetric anisotropic exchange of about 10% is necessary to account
for the basic experimental observations, including the 3D saturation field, and
which in turn might stabilize a triatic (three-magnon) multipolar phase.
