diff --git a/src/sage/algebras/clifford_algebra.py b/src/sage/algebras/clifford_algebra.py
index dda85132dc7..88046a341a7 100644
--- a/src/sage/algebras/clifford_algebra.py
+++ b/src/sage/algebras/clifford_algebra.py
@@ -2776,6 +2776,78 @@ def __richcmp__(self, other, op):
          # remaining case <
         return contained and not contains
 
+    def __mul__(self, other):
+        """
+        Return the product of ``self`` with ``other``.
+
+        .. WARNING::
+
+            If ``self`` is a right ideal and ``other`` is a left ideal,
+            this returns a submodule rather than an ideal.
+
+        EXAMPLES::
+
+            sage: E.<a,b,c,d> = ExteriorAlgebra(QQ)
+
+            sage: I = E.ideal([a + 1], side="left")
+            sage: J = I * I; J
+            Left Ideal (2*a + 1, a, b, c, d, a*b, a*c, a*d, 2*a*b*c + b*c, 2*a*b*d + b*d,
+                        2*a*c*d + c*d, a*b*c, a*b*d, a*c*d, b*c*d, a*b*c*d)
+             of The exterior algebra of rank 4 over Rational Field
+            sage: J.groebner_basis()
+            (1,)
+            sage: I.gen(0)^2
+            2*a + 1
+
+            sage: J = E.ideal([b+c])
+            sage: I * J
+            Twosided Ideal (a*b + a*c + b + c) of The exterior algebra of rank 4 over Rational Field
+            sage: J * I
+            Left Ideal (-a*b - a*c + b + c) of The exterior algebra of rank 4 over Rational Field
+
+            sage: K = J * I
+            sage: K
+            Left Ideal (-a*b - a*c + b + c) of The exterior algebra of rank 4 over Rational Field
+            sage: E.ideal([J.gen(0) * d * I.gen(0)], side="left") <= K
+            True
+
+            sage: J = E.ideal([b + c*d], side="right")
+            sage: I * J
+            Twosided Ideal (a*c*d + a*b + c*d + b) of The exterior algebra of rank 4 over Rational Field
+            sage: X = J * I; X
+            Free module generated by {0, 1, 2, 3, 4, 5, 6, 7} over Rational Field
+            sage: [X.lift(b) for b in X.basis()]
+            [c*d + b, -a*c*d + a*b, b*c, b*d, a*b*c, a*b*d, b*c*d, a*b*c*d]
+            sage: p = X.lift(X.basis()[0])
+            sage: p
+            c*d + b
+            sage: a * p  # not a left ideal
+            a*c*d + a*b
+
+            sage: I = E.ideal([a + 1], side="right")
+            sage: E.ideal([1]) * I
+            Twosided Ideal (a + 1) of The exterior algebra of rank 4 over Rational Field
+            sage: I * E.ideal([1])
+            Right Ideal (a + 1) of The exterior algebra of rank 4 over Rational Field
+        """
+        if not isinstance(other, ExteriorAlgebraIdeal) or self.ring() != other.ring():
+            return super().__mul__(other)
+
+        if self._homogeneous or other._homogeneous or (self.side() == "left" and other.side() == "right"):
+            gens = (x * y for x in self.gens() for y in other.gens())
+        else:
+            gens = (x * t * y for t in self.ring().basis() for x in self.gens() for y in other.gens())
+        gens = [z for z in gens if z]
+
+        if self.side() == "right" and other.side() == "left":
+            return self.ring().submodule(gens)
+
+        if self.side() == "left" or self.side() == "twosided":
+            if other.side() == "right" or other.side() == "twosided":
+                return self.ring().ideal(gens, side="twosided")
+            return self.ring().ideal(gens, side="left")
+        return self.ring().ideal(gens, side="right")
+
     def groebner_basis(self, term_order="neglex"):
         r"""
         Return the reduced Gröbner basis of ``self``.