If I have an isolated composite system which is dependent on several extensive properties (say $U_i, V_i$ and $N_i$ for each subsystem $i$) then a quasi-static locus is just a continuous curve through $U_1-V_1-N_1-U_2-V_2-N_2-\cdots-U_n-V_n-N_n$-space. Moving along this curve means infinitesimally slowly changing the parameters $U_i,V_i$ and $N_i$ so the system moves from one point to the next.
If at any point moving along the locus would cause the entropy of the whole system to drop, then we would not be able to traverse that part of the curve. I'm asking about the opposite problem: if the the entropy along a locus is non-decreasing, can it always be traced out by some physical process?
I realise that no quasi-static process is actually physically realisable - I'm talking about following the curve arbitrarily closely by varying boundary conditions alone.
Another way of posing the questions is: from any particular point in phase space, is it physically possible to move in any direction (within phase space) in which the entropy is not decreasing. If this were true then by taking many small steps we could follow any locus arbitrarily closely.