In contrast, Colombeau 92 defines algebra structures on distributions, now called Colombeau algebras (see Gratus 13 for review) which are globally defined, but at the cost that

they are not uniquely defined,

they do not restrict to the usual pointwise product on all functions.

Briefly: Colombeau considers sequences of functions that converge to distributions (weakly) and defines the product of two distributions as the product of the sequences. This product is not independent of the chosen sequences, which means that the level of abstraction achieved by distribution theory is abandoned.

References

Jean François Colombeau: Multiplication of distributions. A tool in mathematics, numerical engineering and theoretical physics., Springer 1992 (ZMATH entry)

Jonathan Gratus, Colombeau Algebra: A pedagogical introduction (arViv:1308.0257)

Hans Vernaeve, Algebras of generalized functions and Nonstandard analysis, 2008 (pdf)