How are thermodynamic equilibrium processes physically possible? If a system is in thermodynamic equilibrium then the properties of that system are uniform  throughout space and time. As in, over time the properties do not show much bulk evolution. Under this definition, how is a thermodynamic reversible process possible? if a system is truly uniform throughout then it would be impossible to bring out change, and, if we tried to bringout change by changing the various parameters of the substance then we would, by definition, break the thermodynamic equilibrium.
So, how is it possible for statements such as "there always exists a reversible path between two states consisting of consecutive equilibrium states" be true at all? or is it all in reality, an approximation?
Further notes: This consecutive equilibrium picture would require our systems to pass through some discrete equilibrium states, but, from what I know, nature changes continuously.
 A: 
if a system is truly uniform throughout then it would be impossible to
bring out change,

Therein lies the fact that all real processes are irreversible. The process would in theory take an infinite amount of time to perform because it has to be carried out infinitely slowly in order that the system is constantly in equilibrium with the surroundings.
Real processes occur due to disequilibrium. Pressure disequilibrium. Temperature disequilibrium,  chemical disequilibrium, etc.. For example, heat is energy transfer due solely to temperature difference (temperature disequilibrium). Yet for a process to approach reversibility, the temperature difference difference must approach zero which means the rate of heat transfer must approach zero.

Further notes: This consecutive equilibrium picture would require our
systems to pass through some discrete equilibrium states, but, from
what I know, nature changes continuously.

You are correct. That's because in order for the process to be carried out quasi-statically it is necessary to deliberately control the process so that equilibrium is reestablished at each point along the path before incrementally moving to the next point. It's like stop, wait for equilibrium, take an infinitesimal step, repeat. The longer the stop time compared to the time to take the next step, the "smoother" the process overall.
Bottom line: Reversible processes are idealizations to establish an upper limit to what is possible. All real processes are irreversible.
Hope this helps.
