Step 1: Understanding the Concept:
The question describes a situation where two parallel lawsuits are proceeding between the same parties on the same issue. The law has a mechanism to prevent this to avoid conflicting judgments and multiplicity of proceedings. This principle is known as \textit{res sub judice}, which means "the matter is under judgment."
Step 2: Detailed Explanation:
The relevant sections of the Code of Civil Procedure (CPC), 1908 are:
\begin{itemize}
\item Section 9: States that civil courts have the jurisdiction to try all suits of a civil nature unless they are expressly or impliedly barred.
\item Section 10 (Stay of suit): This section directly incorporates the doctrine of \textit{res sub judice}. It mandates that no court shall proceed with the trial of any suit in which the matter in issue is also directly and substantially in issue in a previously instituted suit between the same parties, litigating under the same title, where such suit is pending in the same or any other court in India having jurisdiction. The description in the question is a perfect match for the conditions laid out in Section 10. The later suit is not dismissed, but its trial is stayed.
\item Section 11 (Res Judicata): This section applies when a matter has already been \textit{finally decided} by a competent court. It bars the trial of a subsequent suit on the same matter. The key difference is that for S.10 the previous suit is pending, while for S.11 it is decided.
\item Section 12: This section bars further suits where a plaintiff is precluded by rules from instituting a further suit in respect of a particular cause of action.
\end{itemize}
Step 3: Final Answer:
The provision that governs this situation and leads to the stay of the subsequently filed suit is Section 10 of the CPC.