Foundations
Let's first define a process as a change in a system from an initial state to a different final state.
The term "possible" is perhaps one point of confusion. Any process can be proposed to be a possible process. Only certain such proposed processes are however spontaneous. They will occur with no external input. Other processes are not spontaneous.
The second important point of clarification distinction between $S_{gen}$ and $\Delta S$ for any process. The $\Delta S$ for any process in any one location (control volume, system, or surroundings) includes two terms. One is the reversible entropy change $\Delta S_{rev}$ of the process. The other is the irreversible entropy change $\Delta S_{irr}$ of the process. In engineering applications, the latter term is also called the entropy generation $S_{gen}$. A further discussion of the meaning of these two terms is found at the answer for this question.
Now to the key point. Irreversible entropy generation is always a positive value. We do not define an irreversible process for a given control volume or system or surroundings to generate a negative irreversible entropy. An insight to this is also that we use $S_{gen}$ rather than $\Delta S_{irr}$ specifically because the latter form with a $\Delta$ might suggest that irreversible processes could generate a positive or negative irreversible change, and this is not true.
The entropy criteria that is used to determine whether a process is spontaneous must use the TOTAL entropy change $\Delta S_T$. This is typically called the entropy change of the universe. It is the sum of the entropy change of the system and the surroundings $\Delta S_T = \Delta S_{sys} + \Delta S_{surr}$. Each term for the system or surroundings $\Delta S$ has both reversible entropy change and irreversible entropy change (generation).
When the total entropy change of any proposed process is positive, the process will be spontaneous as proposed. When the system is left to its own, the preferred direction is for it to go from the initial state to the proposed final state. Spontaneous processes are not associated with the direction of heat flow to/from the system. However, they are associated with the direction of heat flow from hot to cold. Ice melts spontaneously at room temperature because heat flows into it. The entropy of the ice increases as it goes to water. Water freezes at temperatures below the freezing point because heat flows out of it. The entropy of the water decreases as it goes to ice. In both cases, the total entropy change of the universe is positive. In both cases, heat spontaneously flows from hot to cold temperature.
When the total entropy change of a proposed process is zero, the process will be at equilibrium at all times throughout the process. The entropy exchanged between the system and the surroundings are exactly equal and opposite. The proposed change from the process will not occur spontaneously. However the process can be driven to occur, and in this case, the process follows a reversible path.
Finally, when the total entropy change is negative, the process will not be spontaneous as proposed. In this case, the inverse process will be spontaneous.
In summary, during any proposed (and therefore hypothetically "possible") process, each entropy component for the system and surroundings can be positive, negative, or zero. The SUM defines whether the proposed process is spontaneous or not.
Specific to Your Example
The example you give has three locations: A, B, and C. In the picture, location C is common to A and B. We would intuitively define C as the surroundings to A and B. For us to state that a process that occurs in this universe is spontaneous, the total entropy change of this universe of system A + system B + surroundings C must be greater than zero. The entropy changes for the separate processes that each occur in system A, system B, and surroundings C can individually be positive, negative, or zero. When the sum is zero, the three locations are in equilibrium. When the sum is negative, the proposed process is not spontaneous as written.
The specific example you post ends with only a sum of three $S_{gen}$ terms. All of the reversible entropy change terms disappear. The sum of the three irreversible terms is the total entropy change of the universe for the process that involves the three locations. The sum must be positive. It is indeed positive in this case because the restriction on any one $S_{gen}$ is that it must be defined as positive when an irreversible process occurs.
Summary
The sign of the entropy change for a process that occurs in a system or in the surroundings is not a metric of whether the process is or is not "possible". It is a sign of the direction of heat flow in or out. It is a sign of whether the system has an increase or a decrease in order.
The sign of the total entropy change of the universe is not a metric of whether a process is or is not "possible". The technical statement is instead precisely that total entropy change is a metric of whether the process is spontaneous as it is proposed.
Entropy change of a defined control volume includes both reversible and irreversible values. The former is $\Delta S_{rev} = \int \delta q_{rev}/T$. The latter is $\Delta S_{irr} \equiv S_{gen}$. The sign of $S_{gen}$ is always positive by definition.
Further Insights
We have an easier time to determine whether a process is or is not spontaneous when we use other thermodynamic state functions instead of entropy. For example, at constant temperature and pressure, we use the Gibbs energy $\Delta G$. We only need to consider the change of the system not of the universe, and the spontaneity criteria is $\Delta G_{sys,T,p} < 0$.
When a proposed process is not spontaneous as written, that does not mean it is not possible. We may have to force the process to go in the reverse direction. Electroplating is an example where we force the process by adding "other work" (electrochemical energy) to the system.