I'm not an expert, but there are variants of LTE and 5G (LTE-U and 5G NR-U) that support operating in the unlicensed 5Ghz band.
That said, I'm not sure a commercial handset would connect to anything in that range. You could probably get an SDR-based UE (UE is telecom-speak for "thing that connects to a cell network") to work.
SDR-based UEs work for 4G right now (see srsLTE with the srsUE), but I'm not aware of an open source one that supports 5G standards yet.
There are unlicensed variants of LTE/5G, but worth noting that these often are designed to use the unlicensed spectrum as supplemental downlink (i.e. License Assisted Access (LAA) in LTE), rather than actually allowing for uplink and downlink in the unlicensed spectrum.
The reason for that is simple politics - mobile operators are powerful lobbyists, and part of a powerful standards group (GSMA), and don't really fancy the idea you should be able to run what a customer will perceive as a mobile network without significant investment in spectrum (a finite resource).
LTE-U and LAA won't let you run a network by yourself just on unlicensed spectrum. Multefire should, but is quite rare for the obvious political reasons outlined above.
Oops sorry. Was referring to the above about mobile operator politics, and lobbying power. Mobile operators like to think of themselves as the only ones owning and operating services on IMT spectrum.
The idea of being able to deliver a full mobile service to mobile phones is "sellable" - operators like to think of themselves as being the only ones able to provide the service people expect to a mobile handset. Then they can bundle handsets with service provision.
If you are interested in this topic, it's worth looking at the fraught relationship between mobile operators and WiFi. Operators historically rejected WiFi from handsets early on - WiFi was a rival to their high-price, high-margin mobile data services. Until perhaps the mid-3G days, when the idea of using WiFi for offload, due to the limited spectrum for mobile data started to become a tempting idea for operators.
Even that's controversial - the "enterprise" market around WiFi wasn't hugely keen on that either, since they felt the mobile operators were just trying to snap up and freeload on the license-exempt spectrum for extra capacity, and use up (finite) WiFi spectrum capacity while providing an operator-badged service.
In my view, Multefire hasn't taken off due to the general high complexity of the tech stack people would need to understand to use it (fine for me with a telecoms engineering background, less fine if you are from the pure IT world and just want something quick - it's a lot more effort and complexity than setting up a couple of WiFi APs), and the lack of pressure from mobile operators to support it. Handset support for features comes from operator demand/desire. Multefire isn't something handset makers will add, unless operators demand it. Absent that, it risks alienating or upsetting them, by opening up the handsets to competition, and since operators are the main route to market for your handsets, market dictates the rules...
LTE-U is commercially available and currently in use on a limited scale for wireless internet service. The main (and almost only) manufacturer of base stations and clients is a company called Baicells.
While LTE-U does have some advantages over WiFi for WISP use (e.g. better handling of a large number of clients per AP), the higher cost (around ~$9k for an LTE-U eNodeB compared to as low as $500 for a WISP-grade 5GHz WiFi AP) and scarcity of vendors means that it's not very commonly used. We'll see if that changes over time, but WiFi has a huge amount of inertia in that space so it seems difficult for LTE-U to catch up considering that the power levels are limited to such an extent that it doesn't have a huge range advantage over WiFi with good antennas.
In fact, just looking at Baicells it seems that they offer fewer models for LTE-U than they used to, so they may be finding that sales aren't enough to keep up the product line.
Apparently, there are overlaps between common LTE bands and both ham radio bands and unlicensed (... low-power) ones... the above page states that any device with the right bands will work (... and they even got them working presumably?)
I remember in the mid 2000s talking with a ham operator. The bands didn't overlap, but they were interleaved - the radios could access bands reserved for cell calls. If you snipped the right resistor off the right radio, then you could listen in on cellphone conversations.
Newer iPhones support band 42 TDD, which is 3400 to 3600 MHz. The amateur radio 9 cm band in the US is 3300 to 3500 MHz, so you could have a setup contained in single ham band. However, the open source eNodeB products typically don't support TDD.
This is the CBRS band, and the idea behind CBRS is that people should be able to gain access to it on an ad-hoc basis by making a request for it, when it isn't being used for marine radar on the US coast.
I believe OpenAirInterface can handle TDD, although it was firmly "research grade" code last time I looked at it.
The positive from CBRS is that it should (or at least is intended to) spawn a new generation of lower cost small cell base stations, using this band, and speaking the CBRS "protocol" for spectrum access coordination. And that has potential to help reduce prices of radio equipment.
Handset compatibility is coming on this band quicker because some existing mobile operators have purchased PALs (priority access licenses) for CBRS spectrum, and intend to use this for some extra capacity.
P.S. just as a very minor technical correction, CBRS is defined for band 48, rather than 42, although with some overlap. B48 is 3550 to 3700 MHz, while B42 is 3.4 to 3.6 GHz as you said. Therefore when looking for devices, it's best to look for B48 (although at a push, if you're doing your own R&D, B42 will be fine for use in the lower 50 MHz section of the band).
Yeah, I'm not an expert either, but there are initiatives to this as well. One such initiative is Nokia Kuha: https://www.kuha.io/
The problem is exactly what you described: there are currently no commercial UE or base stations. So yeah, I suppose one is supposed to use SDR for the 5Ghz band.
It seems that this is aimed exclusively at operators that already have the infrastructure to issue SIM cards et al, and communities that are barred from deploying their own infrastructure but willing to pay for it. A weird use case but clearly aimed at US and AU markets I'd guess.
You need a community that will want to front the cost of laying fiber, and then instead of distributing or starting a small ISP/coop will give that new termination to a telecom giant who will plug this device into their existing billing network to provision their new subscribers with access under the existing plans costs.
That said, I'm not sure a commercial handset would connect to anything in that range. You could probably get an SDR-based UE (UE is telecom-speak for "thing that connects to a cell network") to work.