In many languages, number and gender are somewhat dependent, somewhat independent. C.f. French il, ils, elle, elles.
Naturally, sometimes there will be conflicts in marking. French is the standard example as far as this goes, and the basic mechanism is, I guess, fairly common: if there's even a single man in a group, the whole group as an entity is masculine.
In Indo-European languages, number and gender is fusional (also with case), e.g. in historical Swedish, -or is +fem, +plur (, +nom); -ar is +masc, +plur (, +nom), -n is +neut, +plur (, +nom/acc).
What if we entirely separate the number and gender markers into a more purely agglutinating system. (NB: in modern Swedish, there is almost a hint at that, if we consider -r a plural marker and the preceding vowel a gender marker.)
Let's start out with not having any zero-marked gender, or at least having the zero-marking only pop up in very limited contexts. For this part of the post, I entirely ignore ideas like case, definiteness, etc.
The setup will be thus:
On nouns, some gender may be zero-marked, and the number is zero-marked for singulars. Determiners and pronouns may consist of as little as the gender and number marker with no root, although most pronouns (such as indefinites, various demonstratives, etc) do have roots. The * on gender and number at nouns signify that they're not necessarily always explicitly marked - some nouns may have inherent gender, or possibly, some gender is zero-marked in the noun morphology.
Now for the interesting parts: constructions where the gender or the number is omitted for congruence reasons.
Disjunctions are an obvious contender for such constructions:
Is--[sg] Eve or Peter responsible--[sg] for this.
Here, we could actually consider a meaning distinction encoded in the congruence on the adjective: if the number is unspecified, we leave it open that the adjective is plural - and thus that they both are responsible. Imagine, however, this type of construction:
Is-[masc]-[dual] Peter, John or Albert responsible-[masc]-[dual] for this?
Are we now asking which two out of the three that are responsible?
One more extreme approach could be having disjunctions block all gender marking, such that
Is--[sg?] Peter or Evan responsible--[sg?]
is the only permissible construction. I am a bit partial to that idea myself - I like having the structure per se be the triggering factor instead of the actual gender difference.
2. Indefinite pronouns
Sometimes, we know something about otherwise indefinite actants. E.g. "I saw someone outside the door" - sometimes, you did see enough to be able to specify further. Obviously, sometimes you saw more than one person; sometimes you may be unsure if the several instances of seeing people actually were the same person in slightly different times. Sometimes you have a good guess as to the sex of a person. Sometimes, you may think you've seen one or several men, but you're sure they're all men.
So, in a gender-centered grammatical system, the utility of being able to specify additional optional information - but potentially also omitting it depending on the available knowledge - should be clear.
A distinction between "multiple persons, with several genders in the group" vs. "multiple persons, I was unable to distinguish their genders from the information I got" is possible, but I don't really prefer that kind of system in my own sketches of conlangs, because, well, introducing such a meta-distinction is just not how I roll with under- or overspecifying information in languages in this blog.
3. Non-conjunction-like grouping
In many languages, "and" and "with" basically are not strongly distinguished. In languages that do, however, we could consider a system whereby the number fails to agree with whoever it really agrees with:
I is--[pl] playing music with them
4. Different rules for gender markers and plural markers?
We could also consider a situation whereby the scoping rules for the two markers behave differently over conjunctions, etc, so that
I-masc or they-fem will--[plur] win this game
they-masc or she will--[plur] win this game
In this case, the scoping rule for gender is that gender-disagreement leads to no marking, but plural marking outranks singular marking and always wins if possible.
Another possibility could be that any coordination will trigger plural marking, but congruent gender will permit gender marking:
I-masc or she will--[plur] win
he or she will--[plur] win
Tim or Tom will-[masc]-[plur] win
We could also consider rules like "leftmost number but rightmost gender takes precedence for marking".