Short answer. Yes, but only in that specific situation. Section 10 of Rule 6 lets a defendant file a rejoinder to a reply, but only if the rejoinder is based solely on an actionable document attached to that reply. A rejoinder is not available simply because you disagree with what the reply argues.
What the law says
the plaintiff may file a reply only if the defending party attaches an actionable document to his or her answer
Rule 6, Section 10 — Reply. Read the full provision →
What the law says
A reply is a pleading, the office or function of which is to deny, or allege facts in denial or avoidance of new matters alleged in, or relating to, said actionable document.
Rule 6, Section 10 — Reply. Read the full provision →
What the law says
In the event of an actionable document attached to the reply, the defendant may file a rejoinder if the same is based solely on an actionable document.
Rule 6, Section 10 — Reply. Read the full provision →
Why a reply exists at all
A reply is not a routine pleading a plaintiff can file whenever they want the last word. Under Section 10 of Rule 6, the plaintiff may file a reply only if the defending party attaches an actionable document to his or her answer. It only comes into play once your own answer attached the kind of document that can be sued or defended on directly, and its purpose is narrow: the office or function of which is to deny, or allege facts in denial or avoidance of new matters alleged in, or relating to, said actionable document.
The rejoinder answers the same, narrow question
The rule closes the loop with a matching, equally narrow response for you. In the event of an actionable document attached to the reply, the defendant may file a rejoinder if the same is based solely on an actionable document. So if the plaintiff's reply attached its own actionable document, you may file a rejoinder — but the rejoinder has to be based solely on that document, not a general reply to whatever else the plaintiff argued.
What this means for your rejoinder
Confirm first that what the plaintiff attached is genuinely an actionable document — something the claim or defense is founded on, not just an exhibit offered for context. If it is, your rejoinder should confine itself to denying it, or to alleging facts in denial or avoidance of it, mirroring how the reply itself is limited to responding to the actionable document you attached to your answer. A rejoinder that strays beyond that document goes beyond what Section 10 authorizes. If the plaintiff's reply raised points that have nothing to do with the attached document, those points are not something the rule gives you a rejoinder to answer at all.