What if in a Presidential election..........

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after recounts every state and DC comes in tied between the Republican and the Democrat?

Impossible I know, but if that were the case, what happens?

I'm assuming each state has a chief elections officer who votes only to break a tie.

I doubt this and I doubt law is even providing such an option, knowing odds are indeed impossible. Yet, the question is interesting and I'd love to see some of our in-house law scholars to make an opinion.

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I don't think it would be legal for a chief elections officer to cast a tie-breaking vote.

It would certainly violate "one man - one vote" principle, since said officer certainly already casted his vote as a citizen.

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Each state's electors would then be determined by a coin flip.  That would be an odd-looking electoral map.

William Jennings Bryan lives

Well, I just flipped a coin for each state and, by accident, the map looks pretty consistent.

R: 279
D: 259

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I did the flip for all states again, and map looks much more bizzare this time.

WHAT THE...

It's again R: 279 and D: 259

THIS COIN IS CURSED!

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Damn you, cursed Polish Zloty.

I'll use a Lithuanian Litas now.

D: 276
R: 262

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20 Lithuanian Cents went Democratic.

D: 332
R: 206

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Federal law allows each state + DC to appoint their apportioned electors any way they choose, and this right extends to the settling of disputes in the popular vote.  The deadline for resolving these determinations, however, is six days prior to the meeting of the electors to cast their votes ("the first Monday after the second Wednesday in December").

Your assumption that there is a chief elections officer in each state who only votes in the case of a tie isn't accurate, and I don't know of any state that does so.  Some states might hold a quick revote (if they even have time to do so after the recount deliberations and inevitable lawsuits have concluded), while others will likely have coin flip-type solutions.  It all depends on state law.

Now, who wants to research 51 laws on how to break a tie in the statewide/districtwide popular vote? 

I wonder if there are states with no established provisions for such a case.

In Poland, if election results in a tie, the law provides a randomization. When my dad sat at the  precinct commission during local elections of 2002, he had to perform a randomization for our councilman. Unfortunately, it resulted in our preferred candidate's defeat

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1 złoty, this time, polish coinage went Democratic.

But Georgia stays solid Republican

D: 282
R: 256

How do you guys have the patience to flip a coin 51 times ?

Contraty to what you may think, tt doesn't take long.