Conventionalism and Realism are often presented as alternatives (for example, I recently heard a talk about whether Frege should be understood as a realist or a conventionalist about number). But (at least on my own best understanding of what `conventionalism' might be) it's not at all clear that this is the case.
I'm tempted to understand realism and conventionalism as follows, in which case (I am going to argue) the two are perfectly compatible.
You are a realist about Xs iff you think there really are some Xs.
You are a conventionalist about Xs iff you think that we can reasonably address boundary disputes about just what is to count as an X, or what properties Xs are supposed to have by imposing arbitrary conventions.
Here's an example. I think there really are living things. But I don't think the distinction between living and non-living things is such an incredibly natural kind that much would be lost by stipulating some slight re-definition of "alive" that clearly entails viruses are/aren't "alive". Hence, (by the above definition) I'm both a realist and a conventionalist about living things.
Maybe compatibility between realism about Xs and conventionalism about certain facts about Xs only applies conventionalism with regard to tiny boundary disputes about the extension of the concept X? But here's another example where the extension of X will be completely different depending on what stipulation we make.
I'm a realist about human bodies, in that I think that there are indeed human bodies. But should human bodies be identified with *open* or *closed* sets of space time points? This issue, is (just like the viruses question above) one that it seems perfectly natural to settle by stipulation.
Thus, I don't buy the argument that Frege's willingness to allow some questions about what the numbers are to be determined by convention (assuming, as the speaker suggested, he was indeed so willing) shows that he's an anti-realist about about number in anything like the ordinary sense of the term.
[edit: To put the point another way - you can be a realist about the all items that potentially count as numbers but think it's vague which things exactly do count as numbers.
Taking the extension of a concept to be somewhat arbitrary/conventional doesn't require thinking that the objects which are candidates to fall under that concept are somehow unreal]