Monotone σ-complete groups with unbounded refinement

The real line ℝ may be characterized as the unique non-atomic directed partially ordered abelian group which is monotone σ-complete (countable increasing bounded sequences have suprema), has the countable refinement property (countable sums ${\displaystyle {\sum }_m{a}_m={\sum }_n{b}_n}$ of positive (possibly infinite) elements have common refinements) and 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 the above listed axioms except linear ordering.