If a husband and wife were co-workers, would the wife need to be obedient in the workplace?

If the wife is the boss then the husband must follow her official orders. That is because cheating is haram as stated in the following:

Cheating is haraam, whether it is in buying and selling, or in exams, or any other matter, because of the general meaning of the words of the Prophet (blessings and peace of Allah be upon him): “Whoever cheats is not one of us.”Narrated by Muslim, 102 from the hadeeth of Abu Hurayrah (may Allah be pleased with him)

 

https://islamqa.info/en/175744

. It seems possible that conflicts of interest could arise,

That's haram big time as stated in the following verse:

Sahih International: And do not approach the orphan's property except in a way that is best until he reaches maturity. And give full measure and weight in justice. We do not charge any soul except [with that within] its capacity. And when you testify, be just, even if [it concerns] a near relative. And the covenant of Allah fulfill. This has He instructed you that you may remember.

 

Verse (6:152) - English Translation

If a husband and wife were co-workers, would the wife need to be obedient in the workplace?

If the wife is the boss then the husband must follow her official orders. That is because cheating is haram as stated in the following:

Cheating is haraam, whether it is in buying and selling, or in exams, or any other matter, because of the general meaning of the words of the Prophet (blessings and peace of Allah be upon him): “Whoever cheats is not one of us.”Narrated by Muslim, 102 from the hadeeth of Abu Hurayrah (may Allah be pleased with him)

https://islamqa.info/en/175744

. It seems possible that conflicts of interest could arise,

That's haram big time as stated in the following verse:

Sahih International: And do not approach the orphan's property except in a way that is best until he reaches maturity. And give full measure and weight in justice. We do not charge any soul except [with that within] its capacity. And when you testify, be just, even if [it concerns] a near relative. And the covenant of Allah fulfill. This has He instructed you that you may remember.

Verse (6:152) - English Translation