By Tennenbaum's theorem, in the usual sense of computability for models,
neither addition nor multiplication can be computable in a non-standard model of PA.
Weak version:
Can addition or multiplication become computable if non-equality
only needs to be computably enumerable, rather than computable?
In other words, can there be a countable structure on which [either addition is computable or multiplication is computable] and some quotient of that structure is a non-standard model of PA?
If I understand the proof of the addition version of Tennenbaum's theorem correctly, then for addition to be computable there would need to be a representative r0 of a non-standard number such that {r : r represents the same element as r0} is not computably enumerable.
I'm mainly after an answer to either version, rather than both,
so the following will be quite strong.
Super-Strong version:
Are there
a computable function d : {0,1,2,3,...}2 $\to \mathbb{Q}$
and
binary operations +M and *M on {0,1,2,3,...} (by Tennenbaum, they can't be computable)
such that
composing d with the inclusion from $\mathbb{Q}$ to the real numbers gives a metric
and
the induced metric space is complete
and
there is an algorithm that approximates +M and *M to arbitrary accuracy
and
+M and *M make {0,1,2,3,...} into the non-standard part of a model of PA
?
The standard part could just be put in with distance 1 from everything, so requiring it
to be a non-standard part is stronger than requiring it be a full non-standard model.
Furthermore, for any function and operations witnessing the truth of the Super-Strong version,
[the set of well formed expression using $\hspace{.02 in}0,\hspace{-0.04 in}1,\hspace{-0.04 in}+,\hspace{-0.04 in}*,\hspace{-0.04 in}(\hspace{.02 in},\hspace{-0.03 in})\hspace{.02 in}$ and elements of the metric space]
is a non-standard model in the sense of the weak version,
which both operations are computable in.