There is no problem defining the surreal field without global choice.
One can define it in ZFC and considerably weaker theories, for
example with the hereditary birthday construction of left-sets and
right-sets, and also in other ways.

With global choice, the surreal field No is *largest* in the sense
of model-theoretic universality: all other ordered class fields are
isomorphic to a subfield of No.

One can prove this by means of the usual model-theoretic
back-and-forth argument (really only just the "forth" part),
combined with the fact that the surreal field No is set-saturated
and homogeneous.

That is, given any class field $F$, you use global choice to
well-order $F$, and then build up the embedding $j:F\to\text{No}$
in stages. At any stage, you've embedded part of $F$ into the
surreals, and you consider the next point. By saturation, there is
a surreal number realizing the right type, and you can extend the
embedding one more step.

It is the same argument as showing that the rational line
$\mathbb{Q}$ is a universal countable linear order.

Without global choice, the argument seems to break down, and one
cannot seem to get started. Of course, what is left of the argument
is that even without global choice, the surreal field No remains
universal for all set-sized fields (can one handle well-orderable class fields?). But I don't think it is known whether
there is actually or can be a proper class field that does not
embed into No when global choice fails.

If universality fails, it might not mean that there is some other
"larger" field. Rather, what I would expect is simply that there
are some other ordered class fields that don't embed into No,
perhaps none of them maximal even.

I asked an analogous question about the surreals as a universal
class linear order: Is the universality of the surreal number line
a weak global choice
principle?
Basically, it seems that most of us expect that one really needs global choice for
the universality argument, but to my knowledge we don't yet have any proof of this.

I wonder how the universality of the surreals as a linear order relates to its universality as an ordered field? It has both universality properties under global choice, but can one separate these without global choice?