For the last three games, Austin Romine has been the Yankees starting first baseman. Romine is a career .226/.263/.327 hitter (55 wRC+), and hitting almost exactly to his career averages in 2017 (58 wRC+). He has been okay as a defensive first baseman, but has little experience there.
The Yankees are dead last this season in production at first base:
Austin Romine is not the answer here. His 58 wRC+ is actually worse than the Yankees so far this season (61 wRC+). He is far below replacement level.
Chris Carter is also not the answer. Carter was outrighted to Triple-A as soon as Tyler Austin was ready to go. He was called up before yesterday's game; Romine still played first base. The Yankees have judged that Carter is a worse first baseman than Austin Romine. His 77 wRC+ and horrible defense speak to that claim.
Tyler Austin and Greg Bird are not anywhere near available. Austin's MRI revealed a "high grade hamstring sprain" that will likely linger. Greg Bird's ankle issues are not going away, and he may not play at all this season:
So far, the Yankees have been inexplicably reluctant to tap their replacement players in the high minors. Ji Man Choi has a 124 wRC+ at Triple-A. Mike Ford has a 135 wRC+ at Double-A, and pounded a 218 wRC+ at Triple-A this season. Thus far, the Yankees have shown no interest in calling either up. I'm a Mike Ford supporter, but it doesn't seem to be an option they will consider.
So, we're left with trades. Brian Cashman should be working the phones furiously to find a halfway decent first baseman, any warm body, who is better than Austin Romine or Chris Carter. While it would be nice for Cashman to swing a trade for a Lucas Duda or Eric Hosmer or some other legitimate first-division starting first baseman, he could upgrade the team with much less. I suggested John Jaso on Twitter, but there are plenty of names out there that could pull off a 100 wRC+ with experience at the position. Just a few more examples: Danny Valencia, Tommy Joseph, Adam Lind, Jose Martinez. These guys exist, and wouldn't cost very much.
My overall point: Brian Cashman should be desperate for any upgrade at first base. The Yankees are playing the worse first baseman in the majors right now. They don't seem to like their internal options. Even bad options, replacement-level players, are huge upgrades. Saying "fuck it", throwing your hands in the air and starting Austin Romine every day is something last place teams do, not playoff contenders.