This has probably already been pointed out, but they had one. It was called Xperia Play, and I owned one at one point. It was a “PS Official Product”, and even had the face buttons, D-pad, shoulder buttons, and two sensors that acted as the dual analogue. It was a really cool phone, but the Problem was it used Sony’s own Xperia Play Store, instead of Google Play, which was weird because it was Android OS. Needless to say, it wasn’t remotely successful.
The PlayStation Division wanted nothing to do with Xperia, and the Sony board didn't want Ericsson or Google anywhere close to PlayStation. The Xperia Play was built on crap hardware, and an unsupported store front. SIEWWS never released a single game for the device, nor did any major 3rd parties. The Xperia Play was mildly supported by a small group of start up indie developers. People think the Vita had bad support, the Xperia Play never had any support.
Sony simultaniously destroyed the Handheld and Mobile markets with absolutely terrible decisions between 2011 - 2013. They wasted time putting out the Xperia Play on outdated tech, and gave it zero support in 2011. Then they released the Vita, which had 3G, but no built in Texting and Calling support, and no marketing that it could be used for such things via Skype (2012). Finally, they pulled the 3G Vita, silently killed the Xperia Play, and completely dropped any form of Mobile Gaming support (2013).
This left Sony Mobile with no connection to PlayStation, and just made them a "me too" player in the Android market. Sony's most valuable assets, PlayStation OS, PlayStation Games, PlayStation Network, and PlayStation Store were completely absent in their Mobile devices (2014). They ended adding PS4 Remote Play to the Xperia Z3 in late 2014/ early 2015. Finally PlayStation Video and Vue were added to all Android and iOS devices in 2016. However, PS Now has yet to be released to Android or iOS.
So where does Sony Mobile stand in 2018? The Xperia Line has absolutely nothing to set it apart from any other iOS or Android devices. Limited access to some PlayStation assets are available on all iOS and Android devices. Even if people wanted a Sony Phone, in most places they have to do the research to see which model number is supported in their area and under the carrier they are using or planning to use. Then even after all of that, they will end up with a Phone that is still under supported compaired to Samsung, Google, or Apple devices, with absolutely no special gimick to justify going out of their way, and spending more. All while Sony hemorrhages money.
The fix is simple:
-Roll Sony Mobile into Sony Interactive Entertainment.
-Put the Xperia staff to work on building a PlayStation Phone that runs a Mobile SOC using the same Zen and Navi Architecture as PS5.
-Make PlayStation Phone and PlayStation 5 a unified ecosystem. Same OS, Same Library, a seamless expereince across both devices.
-Make it available through all major carriers.
They won't take on Apple or Samsung for the sales crown, but they don't have to. Even if they sell 1 Million units a year, they will be in a better position than they are now. They don't need Android apps, because they are not aiming to sell to that crowd. They need to aim at the crowd that prefers games over anything else. If a games 1st phone, with access to all of your app needs through a robust browser does not suit your needs, then it is not the device for you.
At the end of the day. This puts Sony's focus 100% on making their platform better. R&D costs are shared between PS5, PSVR2, and PS Phone. The software libarary is shared across PS5, PSVR2, and PS Phone. Sony is no longer supporting:
PLAYSTATION 3 PowerPC CPU/Nvidia GPU/PS3 OS
PLAYSTATION VITA ARM CPU/PowerVR GPU/ PS Vita OS
XPERIA ARM CPU/Qualcomm GPU/ Android OS.
PS5, PSVR2, and PS Phone all run X86 CPU/AMD GPU/PlayStation OS. This would greatly reduce waste/redundancy across the company. Now Sony is selling Sony devices, that continue the relationship beyond the point of sale. They keep users with PS5 and PS Phone engaged in their ecosystem 24/7. They give people the option to enter their ecosystem via Mobile, TV, or VR.
Sony is never going to Compete or be Profitable in Mobile as long as they stay on Android, but they have everything they need to become extremely profitable by unifying behind their own OS and Ecosystem. They will likely never sell 100 million PS Phones a year, but they don't have to. They just have to build an ecosystem that is accessable and profitable. If that ecosystem is 100 million PS5's, 30 Million PS Phone's, and 10 Million PSVR2's, or 60 million PS Phones, 50 Million PS5's, and 30 Million PSVR2's at the end of the day they are all PlayStation users, and Sony is making a profit on whichever combo of devices they buy.