Monotone sigma-complete groups with unbounded refinement

The real line R may be characterized as the unique nonatomic directed partially ordered abelian group which is monotone sigma-complete (countable increasing bounded sequences have suprema), satisfies the countable refinement property (countable sums $\sum_ma_m=\sum_nb_n$ of positive elements $a_m$, $b_n$ have common refinements) and that is linearly ordered. We prove here that the latter condition is not redundant, thus solving an old problem by A. Tarski, by proving that there are many spaces (in particular, of arbitrarily large cardinality) satisfying all above listed axioms except linear ordering