Israel is Visa free to both U.S. and Russian citizens. Upon entry you have 90 days and only need I.D. and say; A ticket with a departure date to prove your intent to vacate within the 90 day cycle.
There are many other countries to go to for a few days/week whatever. Many of them are Visa free also. Poland for one.
But can you return to Israel quickly? Or is there an extended time line you have to follow. Meaning you must stay out for a determined period before re-entry. This relates to this sites info in that a U.S. and Russian citizens can live there verses being apart during a K-1 duration.
We are not rich. There are a number of countries that welcome both citizens but we can't financially live on room service, and hop countries every 88 days.
She's receiving a lot of very good to great job offers that include a room, and some include a separate flat fully furnished as part of the payment she will be receiving. Which is attractive. But as a U.S. guy I don't want to be with her 90 days and then have to leave her for some extended period that defeats the purpose of our plan.
Soon I'll just call up the consulates. But I'm hoping someone has either the information to share. Or guidance to where it is online. I've Googled everything I can think of. And currently I'm saving every nickel for this most unusual future that may take a year. Or the better part of one. In advance; Thanks so very much for any assistance!!!
That Wikipedia page I sent last time looks like a good starting point for answers to this question. See Visa requirements for United States citizens - Wikipedia.
You can also search for any country at travel.state.gov.
If your girl's job offers are in Israel, then your only real question is whether Israel has the 90/180 rule, or just the straight 90 rule. What I call the "90/180 rule" (and that's just my name for it) means you can stay 90 days visa-free in any 180 day period. Or, more simply, if you stay 90 consecutive days you need to be out 90 consecutive days before you return.
I think if I were in your shoes I would just call the U.S. Embassy in Israel to confirm. The contact details are at that travel.state.gov link above.
Then if Israel really has the simple "90 day rule" (stay 90, leave for a day, then return for another 90, etc.) your problem is solved. Every 88 days you take a long weekend trip somewhere nearby to reset the clock.
If Israel's rule is not so simple then you have some decisions to make, mostly about where else you might be able to live.
If doing this is not contingent upon her income at a new job, then you could just alternate between two countries with the 90/180 rule. Such countries include Ukraine, Moldova, Poland, Czech Republic, Lithuania, Latvia, Romania, Hungary, etc.
As for "room service"... Use Airbnb! Seriously, between Uber, Airbnb, and Google Translate the world is now open!
On Airbnb you can rent a bed in a youth hostel in Budapest for $10/night, a mansion in Malibu overlooking the ocean for $2000/night, or anything in between.
And one trick I recommend on Airbnb is to use the filtering to list only "Entire Home" (includes apartments) and only those hosted by "Superhosts". "Superhosts" are those who have hosted many airbnb visitors and have a very high average rating from guests.
By doing that you can easily keep your housing costs under $600-$700 per MONTH. I stayed in just such an airbnb apartment in Kiev for $750/month, and that was as in the middle of the city center as you can get, and it was a HUGE apartment. The premium hotel next door rents rooms starting at over $200/night. Yeah, there's no butler or bellboy at the airbnb, but I can live without that.
So there... Three months in Kiev, three months in Kishinev, Moldova, then if K1 visa still not approved, three more months in Kiev!
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