Most of the time, even if you're playing a variable pitch instrument you're going to be tuning to fixed pitch because you'll have at least one fixed pitch instrument (eg a piano) and if you don't you'll just sound out of tune.
In cases (eg a consort group or string quartet or something) where you're all variable pitch, you'll be tuning to each other and to the scale/key as appropriate to the music and whatever sounds good. You may well sweeten the thirds or widen the fifths a bit etc but that doesn't apply to this question here because you're really not going to see the enharmonic equivalents in the same piece the absolute vast majority of the time for stylistic reasons and if you ever did you would just be tuning to each other to make the vertical incidences sound good rather than thinking consciously of tuning a d-sharp one way and an e-flat another way.
Source: Have a degree and postgrad in music, used to be a professional double bass player[1], spouse has a degree and postgrad in music and teaches at 2 conservatoires in London as well as performing professionally, mostly early music in small consort groups where this sort of tuning thing comes up a lot.
[1] So yeah you can make the standard joke about what do double bass players know about tuning.
I guess it really depends on your instrument, taste, style and the group you are playing with. While studying Cello I actually had a lot of lessons with string quartet where we were analysing the score (harmony) for intonation and it happens quite often in modulations that enharmonic equivalents were used to distinguish whether a chord belongs to the old or the new harmony. And sometimes we really needed to make a difference between an e flat and a d sharp to match an open string or to get a desired tension.
For me the enharmonic equivalent is usually just a totally different harmony, so that is what I tune to. As a result they are quite different notes. I try to do that consciously - also while playing with fixed pitch instruments when possible (like the grandparent comment explained).
Even if playing with a fixed-pitch instrument, it only really sounds out of tune if they're playing the same notes. Which in the styles I play isn't an issue.
So I guess how often this happens in practice varies between styles and eras of music, which would make sense to me. I haven't ever done early music and know nothing about it (other than trombones used to be designed terribly and we now know how to make better ones ;)
Source: Also have a postgrad in music, probably from one of the conservatories your spouse teaches at, and still play trombone professionally.
Most of the time, even if you're playing a variable pitch instrument you're going to be tuning to fixed pitch because you'll have at least one fixed pitch instrument (eg a piano) and if you don't you'll just sound out of tune.
In cases (eg a consort group or string quartet or something) where you're all variable pitch, you'll be tuning to each other and to the scale/key as appropriate to the music and whatever sounds good. You may well sweeten the thirds or widen the fifths a bit etc but that doesn't apply to this question here because you're really not going to see the enharmonic equivalents in the same piece the absolute vast majority of the time for stylistic reasons and if you ever did you would just be tuning to each other to make the vertical incidences sound good rather than thinking consciously of tuning a d-sharp one way and an e-flat another way.
Source: Have a degree and postgrad in music, used to be a professional double bass player[1], spouse has a degree and postgrad in music and teaches at 2 conservatoires in London as well as performing professionally, mostly early music in small consort groups where this sort of tuning thing comes up a lot.
[1] So yeah you can make the standard joke about what do double bass players know about tuning.