In a hypothetical North American setting, assume that a tenure-track candidate whose spouse is also with them during their campus visit. Having lunch with the head of department is common for such a candidate on that day. But, may the candidate's spouse also accompany them during their lunch? Any hints in this regard are appreciated.
I don't think there's an ethical question here, but perhaps an etiquette question. And the answer to that question is: No, I wouldn't suggest this to a hiring Department.
As a cultural matter, every meeting in an interview schedule, including lunch, is considered a business meeting. So lunch isn't a social outing.
In general, the rule is:
During the interview (prior to an offer, as in your question), don't bring your spouse or partner to any events. Many people don't even mention their spouse, because it can unfortunately be a source of discrimination if you have a two-body constraint or you are planning to raise a family, particularly for women candidates.
During the post-offer visit, you can (and are even encouraged to!) bring your spouse or partner to events where they might have a say, or where they might have a vested interest in what's being discussed. This could very well include a meeting with the head of the department if you are planning to chat about, e.g., whether the department holds social events, whether it supports a couple in making housing and job decisions, and general culture of the department.
I have been HoD (math) twice and in my uni, there is no lunch/dinner specifically with the HoD, it's usually members of the hiring committee that attend. Lunches/dinners are most definitely part of the interview process, we're just checking the absence of red flags (e.g., if you start insulting the waitstaff, what would be your attitude towards the janitorial staff be). And also trying to evaluate your capacity to function in a group. A partner would not come to Department Council meetings and thus has no reason to attend that type of event.
Now, there's another reason they should not be present. My uni has a strong spousal hiring program and it extremely important that a candidate not show their cards in that respect prior to being extended an offer. Indeed, although it can be done, it is an immense hassle, in particular for the HoD, to go through the process of spousal hiring. And the hiring committee knows that and also knows that the process is likely to delay the hiring by one, if not two months. A candidate that would come with a two body problem would therefore probably lower their chances of being extended an offer as compared to someone without a two body problem. (The right moment to bring up the issue is after an offer has been made.)
There are of course exceptions: someone hired at a higher level would probably be more open. (Math is a small world.. if you are the prof level and have a spouse also in the field, it's likely that this is known.)
Just as a smallish counterpoint to @Anonymous M's good answer: if the spouse has a significant decision-making role in the candidate's accepting/not a job offer, it would be entirely reasonable to (propose to) include them. In that case, if the dept somehow vetos the spouse's presence, that'd be a bad sign about the atmosphere/politics of the dept.
But, yes, as @Anonymous M correctly observes, this is rarely done, in the U.S., in math depts, in my observation... "for good or for ill" ...
Firstly, there is no automatic lunch with the HoD or anyone else prior to your being offered a position or rejected.
But let's say that there is such an event. And yes, sometimes academics - both junior as well as senior - will use small meals, especially afternoon tea in UK to appraise a candidate in a more relaxed situation. Or maybe just to fit an interview into a busy day's schedule.
What would a spouse be doing at such a lunch ?
Unofficially scouting for a job of their own in the same department ? Such intentions will stick out a mile in the emotional landscape of the lunch. I'd expect a HoD to be quite disgruntled about this outrageous maneuver. There is ample means for an applicant with an academic wife to state their personal situation in their initial letter and depending on the employing university's policy on such things there may be a separate dialogue on that issue.
If the spouse is there to make their other half appear somehow more substantial or to offer emotional and/or moral support in putting across certain views that may or may not be controversial then the question must arise as to why the candidate needs such support in a simple one-on-two situation . . . It's going to be hard for the HoD to avoid the idea that this candidate is somehow inadequate - or at least fears so - as an individual representing themselves. Certainly not the type of person a HoD could feel would be a pillar of strength to the department.
If the spouse imagines that an academic selection process is just like a lot of simple social events, e.g. dinner parties, community group meetings, PTA meetings, political dinners, etc, whereby someone with a spouse is often subtly accorded a higher status than someone without one, then that spouse is making a serious miscalculation about academia. A spouse is a personal support yet never a professional one: one's colleagues fill the latter role.
If the spouse wants to be there because he/she disbelieves what their partner has told them about this job's terms & conditions and wants to hear it from the horse's mouth - well that is a damn bad reflection on the marriage.
Lastly, if the candidate wants their spouse there to neutralize any disadvantage they feel due to the HoD being of the opposite gender, I think the request would evoke a strong reaction where the HoD is a woman. For over a century women academics had to work against the head of not only "male privilege" in the profession but also usually faced all-male interview boards - and on their own. To a woman HoD, any male applicant seeking to have his spouse beside him at any point during the selection process would be seeking a benefit neither sought nor obtainable by woman applicants throughout the history of academia. Your application could hardly go any further. A tenure-track appointment obtained under such circumsances would be worthless: the story would leak out and you'd face a volley of ridiculing looks every day you arrived on campus. No department can afford to draw such a person upon themselves.
From every angle I imagine, I see this notion of yours of bringing a spouse along to an interview-lunch as suicidal in the context of your job application and, given the human tendency to gossip, quite dangerous to any nascent academic career.
If you really want to have a future in academia - moreover in teaching - please please reconsider detachedly (preferably using an impartial senior academic colleague) the appearance of your modus operandi during job applications. Tenure-track candidate selection will for obvious reasons be conservative: the employing department will be obsessed with not hiring an applicant who shows the slightest indicator of uncollegial behavior and will not care a hoot about the risk of missing out on hiring the next Schwinger or Feynman.