Such bases as NaOH, or KOH, which are stronger bases than water, will give OH- ions in water. Also, water itself dissociates to give small amounts of H3O+ and OH-. If a stronger base than H2O is dissolved in water, it will produce OH- and a weak acid.
Assume MA is a very soluble salt, and that A- is a stronger base than H2O.
H2O + M+ + A- -------> M+ + OH- + HA
The OH- is the base now, and HA is a weak acid, having much less effect on pH. Also, the HA might not ordinarily be thought of as an acid; the A- merely accepted a hydrogen ion as it formed a more neutral molecule.