Unconditional consensus is the property of a consensus algorithm for multiple agents, to produce consensus irrespective of the particular time or state at which the agent states are initialized. Under a weak condition, so-called balanced asymmetry, on the sequence (A_t ) of stochastic matrices in the agents states update algorithm, it is shown that (i) the set of accumulation points of states as t grows large is finite, (ii) the asymptotic unconditional occurrence of single consensus or multiple consensuses is directly related to the property of absolute infinite flow of this sequence, as introduced by Touri and Nedic. The latter condition must be satisfied on each of the islands of the so-called unbounded interactions graph induced by (A_t ), defined by Hendricks et al.